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In the previous set of notes, we studied upper bounds on sums such as {|\sum_{N \leq n \leq N+M} n^{-it}|} for {1 \leq M \leq N} that were valid for all {t} in a given range, such as {[T,2T]}; this led in turn to upper bounds on the Riemann zeta {\zeta(\sigma+it)} for {t} in the same range, and for various choices of {\sigma}. While some improvement over the trivial bound of {O(N)} was obtained by these methods, we did not get close to the conjectural bound of {O( N^{1/2+o(1)})} that one expects from pseudorandomness heuristics (assuming that {T} is not too large compared with {N}, e.g. {T = O(N^{O(1)})}.

However, it turns out that one can get much better bounds if one settles for estimating sums such as {|\sum_{N \leq n \leq N+M} n^{-it}|}, or more generally finite Dirichlet series (also known as Dirichlet polynomials) such as {|\sum_n a_n n^{-it}|}, for most values of {t} in a given range such as {[T,2T]}. Equivalently, we will be able to get some control on the large values of such Dirichlet polynomials, in the sense that we can control the set of {t} for which {|\sum_n a_n n^{-it}|} exceeds a certain threshold, even if we cannot show that this set is empty. These large value theorems are often closely tied with estimates for mean values such as {\frac{1}{T}\int_T^{2T} |\sum_n a_n n^{-it}|^{2k}\ dt} of a Dirichlet series; these latter estimates are thus known as mean value theorems for Dirichlet series. Our approach to these theorems will follow the same sort of methods used in Notes 3, in particular relying on the generalised Bessel inequality from those notes.

Our main application of the large value theorems for Dirichlet polynomials will be to control the number of zeroes of the Riemann zeta function {\zeta(s)} (or the Dirichlet {L}-functions {L(s,\chi)}) in various rectangles of the form {\{ \sigma+it: \sigma \geq \alpha, |t| \leq T \}} for various {T > 1} and {1/2 < \alpha < 1}. These rectangles will be larger than the zero-free regions for which we can exclude zeroes completely, but we will often be able to limit the number of zeroes in such rectangles to be quite small. For instance, we will be able to show the following weak form of the Riemann hypothesis: as {T \rightarrow \infty}, a proportion {1-o(1)} of zeroes of the Riemann zeta function in the critical strip with {|\hbox{Im}(s)| \leq T} will have real part {1/2+o(1)}. Related to this, the number of zeroes with {|\hbox{Im}(s)| \leq T} and {|\hbox{Re}(s)| \geq \alpha} can be shown to be bounded by {O( T^{O(1-\alpha)+o(1)} )} as {T \rightarrow \infty} for any {1/2 < \alpha < 1}.

In the next set of notes we will use refined versions of these theorems to establish Linnik’s theorem on the least prime in an arithmetic progression.

Our presentation here is broadly based on Chapters 9 and 10 in Iwaniec and Kowalski, who give a number of more sophisticated large value theorems than the ones discussed here.

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We return to the study of the Riemann zeta function {\zeta(s)}, focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of Notes 2, such upper bounds can lead to zero-free regions for {\zeta}, which in turn lead to improved estimates for the error term in the prime number theorem.

In equation (21) of Notes 2 we obtained the somewhat crude estimates

\displaystyle  \zeta(s) = \sum_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} + O( \frac{|s|}{\sigma} \frac{1}{x^\sigma} ) \ \ \ \ \ (1)

for any {x > 0} and {s = \sigma+it} with {\sigma>0} and {s \neq 1}. Setting {x=1}, we obtained the crude estimate

\displaystyle  \zeta(s) = \frac{1}{s-1} + O( \frac{|s|}{\sigma} )

in this region. In particular, if {0 < \varepsilon \leq \sigma \ll 1} and {|t| \gg 1} then we had {\zeta(s) = O_\varepsilon( |t| )}. Using the functional equation and the Hadamard three lines lemma, we can improve this to {\zeta(s) \ll_\varepsilon |t|^{\frac{1-\sigma}{2}+\varepsilon}}; see Supplement 3.

Now we seek better upper bounds on {\zeta}. We will reduce the problem to that of bounding certain exponential sums, in the spirit of Exercise 33 of Supplement 3:

Proposition 1 Let {s = \sigma+it} with {0 < \varepsilon \leq \sigma \ll 1} and {|t| \gg 1}. Then

\displaystyle  \zeta(s) \ll_\varepsilon \log(2+|t|) \sup_{1 \leq M \leq N \ll |t|} N^{1-\sigma} |\frac{1}{N} \sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n)|

where {e(x) := e^{2\pi i x}}.

Proof: We fix a smooth function {\eta: {\bf R} \rightarrow {\bf C}} with {\eta(t)=1} for {t \leq -1} and {\eta(t)=0} for {t \geq 1}, and allow implied constants to depend on {\eta}. Let {s=\sigma+it} with {\varepsilon \leq \sigma \ll 1}. From Exercise 33 of Supplement 3, we have

\displaystyle  \zeta(s) = \sum_n \frac{1}{n^s} \eta( \log n - \log C|t| ) + O_\varepsilon( 1 )

for some sufficiently large absolute constant {C}. By dyadic decomposition, we thus have

\displaystyle  \zeta(s) \ll_{\varepsilon} 1 + \log(2+|t|) \sup_{1 \leq N \ll |t|} |\sum_{N \leq n < 2N} \frac{1}{n^s} \eta( \log n - \log C|t| )|.

We can absorb the first term in the second using the {N=1} case of the supremum. Writing {\frac{1}{n^s} \eta( \log n - \log|C| t ) = N^{-\sigma} e( - \frac{t}{2\pi} \log n ) F_N(n)}, where

\displaystyle  F_N(n) := (N/n)^\sigma \eta(\log n - \log C|t| ),

it thus suffices to show that

\displaystyle  \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log N) F_N(n) \ll \sup_{1 \leq M \leq N} |\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)|

for each {N}. But from the fundamental theorem of calculus, the left-hand side can be written as

\displaystyle  F_N(2N) \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log n)

\displaystyle - \int_0^{N} (\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)) F'_N(M)\ dM

and the claim then follows from the triangle inequality and a routine calculation. \Box

We are thus interested in getting good bounds on the sum {\sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n )}. More generally, we consider normalised exponential sums of the form

\displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ \ \ \ \ (2)

where {I \subset {\bf R}} is an interval of length at most {N} for some {N \geq 1}, and {f: {\bf R} \rightarrow {\bf R}} is a smooth function. We will assume smoothness estimates of the form

\displaystyle  |f^{(j)}(x)| = \exp( O(j^2) ) \frac{T}{N^j} \ \ \ \ \ (3)

for some {T>0}, all {x \in I}, and all {j \geq 1}, where {f^{(j)}} is the {j}-fold derivative of {f}; in the case {f(x) := -\frac{t}{2\pi} \log x}, {I \subset [N,2N]} of interest for the Riemann zeta function, we easily verify that these estimates hold with {T := |t|}. (One can consider exponential sums under more general hypotheses than (3), but the hypotheses here are adequate for our needs.) We do not bound the zeroth derivative {f^{(0)}=f} of {f} directly, but it would not be natural to do so in any event, since the magnitude of the sum (2) is unaffected if one adds an arbitrary constant to {f(n)}.

The trivial bound for (2) is

\displaystyle  \frac{1}{N} \sum_{n \in I} e(f(n)) \ll 1 \ \ \ \ \ (4)

and we will seek to obtain significant improvements to this bound. Pseudorandomness heuristics predict a bound of {O_\varepsilon(N^{-1/2+\varepsilon})} for (2) for any {\varepsilon>0} if {T = O(N^{O(1)})}; this assertion (a special case of the exponent pair hypothesis) would have many consequences (for instance, inserting it into Proposition 1 soon yields the Lindelöf hypothesis), but is unfortunately quite far from resolution with known methods. However, we can obtain weaker gains of the form {O(N^{1-c_K})} when {T \ll N^K} and {c_K > 0} depends on {K}. We present two such results here, which perform well for small and large values of {K} respectively:

Theorem 2 Let {2 \leq N \ll T}, let {I} be an interval of length at most {N}, and let {f: I \rightarrow {\bf R}} be a smooth function obeying (3) for all {j \geq 1} and {x \in I}.

  • (i) (van der Corput estimate) For any natural number {k \geq 2}, one has

    \displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll (\frac{T}{N^k})^{\frac{1}{2^k-2}} \log^{1/2} (2+T). \ \ \ \ \ (5)

  • (ii) (Vinogradov estimate) If {k} is a natural number and {T \leq N^{k}}, then

    \displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll N^{-c/k^2} \ \ \ \ \ (6)

    for some absolute constant {c>0}.

The factor of {\log^{1/2} (2+T)} can be removed by a more careful argument, but we will not need to do so here as we are willing to lose powers of {\log T}. The estimate (6) is superior to (5) when {T \sim N^K} for {K} large, since (after optimising in {k}) (5) gives a gain of the form {N^{-c/2^{cK}}} over the trivial bound, while (6) gives {N^{-c/K^2}}. We have not attempted to obtain completely optimal estimates here, settling for a relatively simple presentation that still gives good bounds on {\zeta}, and there are a wide variety of additional exponential sum estimates beyond the ones given here; see Chapter 8 of Iwaniec-Kowalski, or Chapters 3-4 of Montgomery, for further discussion.

We now briefly discuss the strategies of proof of Theorem 2. Both parts of the theorem proceed by treating {f} like a polynomial of degree roughly {k}; in the case of (ii), this is done explicitly via Taylor expansion, whereas for (i) it is only at the level of analogy. Both parts of the theorem then try to “linearise” the phase to make it a linear function of the summands (actually in part (ii), it is necessary to introduce an additional variable and make the phase a bilinear function of the summands). The van der Corput estimate achieves this linearisation by squaring the exponential sum about {k} times, which is why the gain is only exponentially small in {k}. The Vinogradov estimate achieves linearisation by raising the exponential sum to a significantly smaller power – on the order of {k^2} – by using Hölder’s inequality in combination with the fact that the discrete curve {\{ (n,n^2,\dots,n^k): n \in \{1,\dots,M\}\}} becomes roughly equidistributed in the box {\{ (a_1,\dots,a_k): a_j = O( M^j ) \}} after taking the sumset of about {k^2} copies of this curve. This latter fact has a precise formulation, known as the Vinogradov mean value theorem, and its proof is the most difficult part of the argument, relying on using a “{p}-adic” version of this equidistribution to reduce the claim at a given scale {M} to a smaller scale {M/p} with {p \sim M^{1/k}}, and then proceeding by induction.

One can combine Theorem 2 with Proposition 1 to obtain various bounds on the Riemann zeta function:

Exercise 3 (Subconvexity bound)

  • (i) Show that {\zeta(\frac{1}{2}+it) \ll (1+|t|)^{1/6} \log^{O(1)}(1+|t|)} for all {t \in {\bf R}}. (Hint: use the {k=3} case of the Van der Corput estimate.)
  • (ii) For any {0 < \sigma < 1}, show that {\zeta(\sigma+it) \ll (1+|t|)^{\max( \frac{1-\sigma}{3}, \frac{1}{2} - \frac{2\sigma}{3}) + o(1)}} as {|t| \rightarrow \infty}.

Exercise 4 Let {t} be such that {|t| \geq 100}, and let {\sigma \geq 1/2}.

  • (i) (Littlewood bound) Use the van der Corput estimate to show that {\zeta(\sigma+it) \ll \log^{O(1)} |t|} whenever {\sigma \geq 1 - O( \frac{(\log\log |t|)^2}{\log |t|} ))}.
  • (ii) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that {\zeta(\sigma+it) \ll \log^{O(1)} |t|} whenever {\sigma \geq 1 - O( \frac{(\log\log |t|)^{2/3}}{\log^{2/3} |t|} )}.

As noted in Exercise 43 of Notes 2, the Vinogradov-Korobov bound leads to the zero-free region {\{ \sigma+it: \sigma > 1 - c \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}; |t| \geq 100 \}}, which in turn leads to the prime number theorem with error term

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + O\left( x \exp\left( - c \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)

for {x > 100}. If one uses the weaker Littlewood bound instead, one obtains the narrower zero-free region

\displaystyle  \{ \sigma+it: \sigma > 1 - c \frac{\log\log|t|}{\log |t|}; |t| \geq 100 \}

(which is only slightly wider than the classical zero-free region) and an error term

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + O( x \exp( - c \sqrt{\log x \log\log x} ) )

in the prime number theorem.

Exercise 5 (Vinogradov-Korobov in arithmetic progressions) Let {\chi} be a non-principal character of modulus {q}.

  • (i) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that {L(\sigma+it,\chi) \ll \log^{O(1)}(q|t|)} whenever {|t| \geq 100} and

    \displaystyle  \sigma \geq 1 - O( \min( \frac{\log\log(q|t|)}{\log q}, \frac{(\log\log(q|t|))^{2/3}}{\log^{2/3} |t|} ) ).

    (Hint: use the Vinogradov estimate and a change of variables to control {\sum_{n \in I: n = a\ (q)} \exp( -it \log n)} for various intervals {I} of length at most {N} and residue classes {a\ (q)}, in the regime {N \geq q^2} (say). For {N < q^2}, do not try to capture any cancellation and just use the triangle inequality instead.)

  • (ii) Obtain a zero-free region

    \displaystyle  \{ \sigma+it: \sigma > 1 - c \min( \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}, \frac{1}{\log q} );

    \displaystyle  |t| \geq 100 \}

    for {L(s,\chi)}, for some (effective) absolute constant {c>0}.

  • (iii) Obtain the prime number theorem in arithmetic progressions with error term

    \displaystyle  \sum_{n \leq x: n = a\ (q)} \Lambda(n) = x + O\left( x \exp\left( - c_A \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)

    whenever {x > 100}, {q \leq \log^A x}, {a\ (q)} is primitive, and {c_A>0} depends (ineffectively) on {A}.

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We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the linear sieve in which the sieve dimension {\kappa} is equal to {1}, which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. A bit more specifically, let {z, D \geq 1} be quantities with {z = D^{1/s}} for some fixed {s>1}, and let {g} be a multiplicative function with

\displaystyle  g(p) = \frac{1}{p} + O(\frac{1}{p^2}) \ \ \ \ \ (1)


\displaystyle  0 \leq g(p) \leq 1-c \ \ \ \ \ (2)

for all primes {p} and some fixed {c>0} (we allow all constants below to depend on {c}). Let {P(z) := \prod_{p<z} p}, and for each prime {p < z}, let {E_p} be a set of integers, with {E_d := \bigcap_{p|d} E_p} for {d|P(z)}. We consider finitely supported sequences {(a_n)_{n \in {\bf Z}}} of non-negative reals for which we have bounds of the form

\displaystyle  \sum_{n \in E_d} a_n = g(d) X + r_d. \ \ \ \ \ (3)

for all square-free {d \leq D} and some {X>0}, and some remainder terms {r_d}. One is then interested in upper and lower bounds on the quantity

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n.

The fundamental lemma of sieve theory (Corollary 19 of Notes 4) gives us the bound

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n = (1 + O(e^{-s})) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (4)

where {V(z)} is the quantity

\displaystyle  V(z) := \prod_{p<z} (1-g(p)). \ \ \ \ \ (5)

This bound is strong when {s} is large, but is not as useful for smaller values of {s}. We now give a sharp bound in this regime. We introduce the functions {F, f: (0,+\infty) \rightarrow {\bf R}^+} by

\displaystyle  F(s) := 2e^\gamma ( \frac{1_{s>1}}{s} \ \ \ \ \ (6)

\displaystyle  + \sum_{j \geq 3, \hbox{ odd}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} )


\displaystyle  f(s) := 2e^\gamma \sum_{j \geq 2, \hbox{ even}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} \ \ \ \ \ (7)

where we adopt the convention {t_j := s - t_1 - \dots - t_{j-1}}. Note that for each {s} one has only finitely many non-zero summands in (6), (7). These functions are closely related to the Buchstab function {\omega} from Exercise 28 of Supplement 4; indeed from comparing the definitions one has

\displaystyle  F(s) + f(s) = 2 e^\gamma \omega(s)

for all {s>0}.

Exercise 1 (Alternate definition of {F, f}) Show that {F(s)} is continuously differentiable except at {s=1}, and {f(s)} is continuously differentiable except at {s=2} where it is continuous, obeying the delay-differential equations

\displaystyle  \frac{d}{ds}( s F(s) ) = f(s-1) \ \ \ \ \ (8)

for {s > 1} and

\displaystyle  \frac{d}{ds}( s f(s) ) = F(s-1) \ \ \ \ \ (9)

for {s>2}, with the initial conditions

\displaystyle  F(s) = \frac{2e^\gamma}{s} 1_{s>1}

for {s \leq 3} and

\displaystyle  f(s) = 0

for {s \leq 2}. Show that these properties of {F, f} determine {F, f} completely.

For future reference, we record the following explicit values of {F, f}:

\displaystyle  F(s) = \frac{2e^\gamma}{s} \ \ \ \ \ (10)

for {1 < s \leq 3}, and

\displaystyle  f(s) = \frac{2e^\gamma}{s} \log(s-1) \ \ \ \ \ (11)

for {2 \leq s \leq 4}.

We will show

Theorem 2 (Linear sieve) Let the notation and hypotheses be as above, with {s > 1}. Then, for any {\varepsilon > 0}, one has the upper bound

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n \leq (F(s) + O(\varepsilon)) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (12)

and the lower bound

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n \geq (f(s) - O(\varepsilon)) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (13)

if {D} is sufficiently large depending on {\varepsilon, s, c}. Furthermore, this claim is sharp in the sense that the quantity {F(s)} cannot be replaced by any smaller quantity, and similarly {f(s)} cannot be replaced by any larger quantity.

Comparing the linear sieve with the fundamental lemma (and also testing using the sequence {a_n = 1_{1 \leq n \leq N}} for some extremely large {N}), we conclude that we necessarily have the asymptotics

\displaystyle  1 - O(e^{-s}) \leq f(s) \leq 1 \leq F(s) \leq 1 + O( e^{-s} )

for all {s \geq 1}; this can also be proven directly from the definitions of {F, f}, or from Exercise 1, but is somewhat challenging to do so; see e.g. Chapter 11 of Friedlander-Iwaniec for details.

Exercise 3 Establish the integral identities

\displaystyle  F(s) = 1 + \frac{1}{s} \int_s^\infty (1 - f(t-1))\ dt


\displaystyle  f(s) = 1 + \frac{1}{s} \int_s^\infty (1 - F(t-1))\ dt

for {s \geq 2}. Argue heuristically that these identities are consistent with the bounds in Theorem 2 and the Buchstab identity (Equation (16) from Notes 4).

Exercise 4 Use the Selberg sieve (Theorem 30 from Notes 4) to obtain a slightly weaker version of (12) in the range {1 < s < 3} in which the error term {|r_d|} is worsened to {\tau_3(d) |r_d|}, but the main term is unchanged.

We will prove Theorem 2 below the fold. The optimality of {F, f} is closely related to the parity problem obstruction discussed in Section 5 of Notes 4; a naive application of the parity arguments there only give the weak bounds {F(s) \geq \frac{2 e^\gamma}{s}} and {f(s)=0} for {s \leq 2}, but this can be sharpened by a more careful counting of various sums involving the Liouville function {\lambda}.

As an application of the linear sieve (specialised to the ranges in (10), (11)), we will establish a famous theorem of Chen, giving (in some sense) the closest approach to the twin prime conjecture that one can hope to achieve by sieve-theoretic methods:

Theorem 5 (Chen’s theorem) There are infinitely many primes {p} such that {p+2} is the product of at most two primes.

The same argument gives the version of Chen’s theorem for the even Goldbach conjecture, namely that for all sufficiently large even {N}, there exists a prime {p} between {2} and {N} such that {N-p} is the product of at most two primes.

The discussion in these notes loosely follows that of Friedlander-Iwaniec (who study sieving problems in more general dimension than {\kappa=1}).

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Many problems in non-multiplicative prime number theory can be recast as sieving problems. Consider for instance the problem of counting the number {N(x)} of pairs of twin primes {p,p+2} contained in {[x/2,x]} for some large {x}; note that the claim that {N(x) > 0} for arbitrarily large {x} is equivalent to the twin prime conjecture. One can obtain this count by any of the following variants of the sieve of Eratosthenes:

  1. Let {A} be the set of natural numbers in {[x/2,x-2]}. For each prime {p \leq \sqrt{x}}, let {E_p} be the union of the residue classes {0\ (p)} and {-2\ (p)}. Then {N(x)} is the cardinality of the sifted set {A \backslash \bigcup_{p \leq \sqrt{x}} E_p}.
  2. Let {A} be the set of primes in {[x/2,x-2]}. For each prime {p \leq \sqrt{x}}, let {E_p} be the residue class {-2\ (p)}. Then {N(x)} is the cardinality of the sifted set {A \backslash \bigcup_{p \leq \sqrt{x}} E_p}.
  3. Let {A} be the set of primes in {[x/2+2,x]}. For each prime {p \leq \sqrt{x}}, let {E_p} be the residue class {2\ (p)}. Then {N(x)} is the cardinality of the sifted set {A \backslash \bigcup_{p \leq \sqrt{x}} E_p}.
  4. Let {A} be the set {\{ n(n+2): x/2 \leq n \leq x-2 \}}. For each prime {p \leq \sqrt{x}}, let {E_p} be the residue class {0\ (p)} Then {N(x)} is the cardinality of the sifted set {A \backslash \bigcup_{p \leq \sqrt{x}} E_p}.

Exercise 1 Develop similar sifting formulations of the other three Landau problems.

In view of these sieving interpretations of number-theoretic problems, it becomes natural to try to estimate the size of sifted sets {A \backslash \bigcup_{p | P} E_p} for various finite sets {A} of integers, and subsets {E_p} of integers indexed by primes {p} dividing some squarefree natural number {P} (which, in the above examples, would be the product of all primes up to {\sqrt{x}}). As we see in the above examples, the sets {E_p} in applications are typically the union of one or more residue classes modulo {p}, but we will work at a more abstract level of generality here by treating {E_p} as more or less arbitrary sets of integers, without caring too much about the arithmetic structure of such sets.

It turns out to be conceptually more natural to replace sets by functions, and to consider the more general the task of estimating sifted sums

\displaystyle  \sum_{n \in {\bf Z}} a_n 1_{n \not \in \bigcup_{p | P} E_p} \ \ \ \ \ (1)

for some finitely supported sequence {(a_n)_{n \in {\bf Z}}} of non-negative numbers; the previous combinatorial sifting problem then corresponds to the indicator function case {a_n=1_{n \in A}}. (One could also use other index sets here than the integers {{\bf Z}} if desired; for much of sieve theory the index set and its subsets {E_p} are treated as abstract sets, so the exact arithmetic structure of these sets is not of primary importance.)

Continuing with twin primes as a running example, we thus have the following sample sieving problem:

Problem 2 (Sieving problem for twin primes) Let {x, z \geq 1}, and let {\pi_2(x,z)} denote the number of natural numbers {n \leq x} which avoid the residue classes {0, -2\ (p)} for all primes {p < z}. In other words, we have

\displaystyle  \pi_2(x,z) := \sum_{n \in {\bf Z}} a_n 1_{n \not \in \bigcup_{p | P(z)} E_p}

where {a_n := 1_{n \in [1,x]}}, {P(z) := \prod_{p < z} p} is the product of all the primes strictly less than {z} (we omit {z} itself for minor technical reasons), and {E_p} is the union of the residue classes {0, -2\ (p)}. Obtain upper and lower bounds on {\pi_2(x,z)} which are as strong as possible in the asymptotic regime where {x} goes to infinity and the sifting level {z} grows with {x} (ideally we would like {z} to grow as fast as {\sqrt{x}}).

From the preceding discussion we know that the number of twin prime pairs {p,p+2} in {(x/2,x]} is equal to {\pi_2(x-2,\sqrt{x}) - \pi_2(x/2,\sqrt{x})}, if {x} is not a perfect square; one also easily sees that the number of twin prime pairs in {[1,x]} is at least {\pi_2(x-2,\sqrt{x})}, again if {x} is not a perfect square. Thus we see that a sufficiently good answer to Problem 2 would resolve the twin prime conjecture, particularly if we can get the sifting level {z} to be as large as {\sqrt{x}}.

We return now to the general problem of estimating (1). We may expand

\displaystyle  1_{n \not \in \bigcup_{p | P} E_p} = \prod_{p | P} (1 - 1_{E_p}(n)) \ \ \ \ \ (2)

\displaystyle  = \sum_{k=0}^\infty (-1)^k \sum_{p_1 \dots p_k|P: p_1 < \dots < p_k} 1_{E_{p_1}} \dots 1_{E_{p_k}}(n)

\displaystyle  = \sum_{d|P} \mu(d) 1_{E_d}(n)

where {E_d := \bigcap_{p|d} E_p} (with the convention that {E_1={\bf Z}}). We thus arrive at the Legendre sieve identity

\displaystyle  \sum_{n \in {\bf Z}} a_n 1_{n \not \in \bigcup_{p | P} E_p} = \sum_{d|P} \mu(d) \sum_{n \in E_d} a_n. \ \ \ \ \ (3)

Specialising to the case of an indicator function {a_n=1_{n \in A}}, we recover the inclusion-exclusion formula

\displaystyle  |A \backslash \bigcup_{p|P} E_p| = \sum_{d|P} \mu(d) |A \cap E_d|.

Such exact sieving formulae are already satisfactory for controlling sifted sets or sifted sums when the amount of sieving is relatively small compared to the size of {A}. For instance, let us return to the running example in Problem 2 for some {x,z \geq 1}. Observe that each {E_p} in this example consists of {\omega(p)} residue classes modulo {p}, where {\omega(p)} is defined to equal {1} when {p=2} and {2} when {p} is odd. By the Chinese remainder theorem, this implies that for each {d|P(z)}, {E_d} consists of {\prod_{p|d} \omega(p)} residue classes modulo {d}. Using the basic bound

\displaystyle  \sum_{n \leq x: n = a\ (q)} 1 = \frac{x}{q} + O(1) \ \ \ \ \ (4)

for any {x > 0} and any residue class {a\ (q)}, we conclude that

\displaystyle  \sum_{n \in E_d} a_n = g(d) x + O( \prod_{p|d} \omega(p) ) \ \ \ \ \ (5)

for any {d|P(z)}, where {g} is the multiplicative function

\displaystyle  g(d) := \prod_{p|d: p|P(z)} \frac{\omega(p)}{p}.

Since {\omega(p) \leq 2} and there are at most {\pi(z)} primes dividing {P(z)}, we may crudely bound {\prod_{p|d} \omega(p) \leq 2^{\pi(z)}}, thus

\displaystyle  \sum_{n \in E_d} a_n = g(d) x + O( 2^{\pi(z)} ). \ \ \ \ \ (6)

Also, the number of divisors of {P(z)} is at most {2^{\pi(z)}}. From the Legendre sieve (3), we thus conclude that

\displaystyle  \pi_2(x,z) = (\sum_{d|P(z)} \mu(d) g(d) x) + O( 4^{\pi(z)} ).

We can factorise the main term to obtain

\displaystyle  \pi_2(x,z) = x \prod_{p < z} (1-\frac{\omega(p)}{p}) + O( 4^{\pi(z)} ).

This is compatible with the heuristic

\displaystyle  \pi_2(x,z) \approx x \prod_{p < z} (1-\frac{\omega(p)}{p}) \ \ \ \ \ (7)

coming from the equidistribution of residues principle (Section 3 of Supplement 4), bearing in mind (from the modified Cramér model, see Section 1 of Supplement 4) that we expect this heuristic to become inaccurate when {z} becomes very large. We can simplify the right-hand side of (7) by recalling the twin prime constant

\displaystyle  \Pi_2 := \prod_{p>2} (1 - \frac{1}{(p-1)^2}) = 0.6601618\dots

(see equation (7) from Supplement 4); note that

\displaystyle  \prod_p (1-\frac{1}{p})^{-2} (1-\frac{\omega(p)}{p}) = 2 \Pi_2

so from Mertens’ third theorem (Theorem 42 from Notes 1) one has

\displaystyle  \prod_{p < z} (1-\frac{\omega(p)}{p}) = (2\Pi_2+o(1)) \frac{1}{(e^\gamma \log z)^2} \ \ \ \ \ (8)

as {z \rightarrow \infty}. Bounding {4^{\pi(z)}} crudely by {\exp(o(z))}, we conclude in particular that

\displaystyle  \pi_2(x,z) = (2\Pi_2 +o(1)) \frac{x}{(e^\gamma \log z)^2}

when {x,z \rightarrow \infty} with {z = O(\log x)}. This is somewhat encouraging for the purposes of getting a sufficiently good answer to Problem 2 to resolve the twin prime conjecture, but note that {z} is currently far too small: one needs to get {z} as large as {\sqrt{x}} before one is counting twin primes, and currently {z} can only get as large as {\log x}.

The problem is that the number of terms in the Legendre sieve (3) basically grows exponentially in {z}, and so the error terms in (4) accumulate to an unacceptable extent once {z} is significantly larger than {\log x}. An alternative way to phrase this problem is that the estimate (4) is only expected to be truly useful in the regime {q=o(x)}; on the other hand, the moduli {d} appearing in (3) can be as large as {P}, which grows exponentially in {z} by the prime number theorem.

To resolve this problem, it is thus natural to try to truncate the Legendre sieve, in such a way that one only uses information about the sums {\sum_{n \in E_d} a_n} for a relatively small number of divisors {d} of {P}, such as those {d} which are below a certain threshold {D}. This leads to the following general sieving problem:

Problem 3 (General sieving problem) Let {P} be a squarefree natural number, and let {{\mathcal D}} be a set of divisors of {P}. For each prime {p} dividing {P}, let {E_p} be a set of integers, and define {E_d := \bigcap_{p|d} E_p} for all {d|P} (with the convention that {E_1={\bf Z}}). Suppose that {(a_n)_{n \in {\bf Z}}} is an (unknown) finitely supported sequence of non-negative reals, whose sums

\displaystyle  X_d := \sum_{n \in E_d} a_n \ \ \ \ \ (9)

are known for all {d \in {\mathcal D}}. What are the best upper and lower bounds one can conclude on the quantity (1)?

Here is a simple example of this type of problem (corresponding to the case {P = 6}, {{\mathcal D} = \{1, 2, 3\}}, {X_1 = 100}, {X_2 = 60}, and {X_3 = 10}):

Exercise 4 Let {(a_n)_{n \in {\bf Z}}} be a finitely supported sequence of non-negative reals such that {\sum_{n \in {\bf Z}} a_n = 100}, {\sum_{n \in {\bf Z}: 2|n} a_n = 60}, and {\sum_{n \in {\bf Z}: 3|n} a_n = 10}. Show that

\displaystyle  30 \leq \sum_{n \in {\bf Z}: (n,6)=1} a_n \leq 40

and give counterexamples to show that these bounds cannot be improved in general, even when {a_n} is an indicator function sequence.

Problem 3 is an example of a linear programming problem. By using linear programming duality (as encapsulated by results such as the Hahn-Banach theorem, the separating hyperplane theorem, or the Farkas lemma), we can rephrase the above problem in terms of upper and lower bound sieves:

Theorem 5 (Dual sieve problem) Let {P, {\mathcal D}, E_p, E_d, X_d} be as in Problem 3. We assume that Problem 3 is feasible, in the sense that there exists at least one finitely supported sequence {(a_n)_{n \in {\bf Z}}} of non-negative reals obeying the constraints in that problem. Define an (normalised) upper bound sieve to be a function {\nu^+: {\bf Z} \rightarrow {\bf R}} of the form

\displaystyle  \nu^+ = \sum_{d \in {\mathcal D}} \lambda^+_d 1_{E_d}

for some coefficients {\lambda^+_d \in {\bf R}}, and obeying the pointwise lower bound

\displaystyle  \nu^+(n) \geq 1_{n \not \in\bigcup_{p|P} E_p}(n) \ \ \ \ \ (10)

for all {n \in {\bf Z}} (in particular {\nu^+} is non-negative). Similarly, define a (normalised) lower bound sieve to be a function {\nu^-: {\bf Z} \rightarrow {\bf R}} of the form

\displaystyle  \nu^-(n) = \sum_{d \in {\mathcal D}} \lambda^-_d 1_{E_d}

for some coefficients {\lambda^-_d \in {\bf R}}, and obeying the pointwise upper bound

\displaystyle  \nu^-(n) \leq 1_{n \not \in\bigcup_{p|P} E_p}(n)

for all {n \in {\bf Z}}. Thus for instance {1} and {0} are (trivially) upper bound sieves and lower bound sieves respectively.

  • (i) The supremal value of the quantity (1), subject to the constraints in Problem 3, is equal to the infimal value of the quantity {\sum_{d \in {\mathcal D}} \lambda^+_d X_d}, as {\nu^+ = \sum_{d \in {\mathcal D}} \lambda^+_d 1_{E_d}} ranges over all upper bound sieves.
  • (ii) The infimal value of the quantity (1), subject to the constraints in Problem 3, is equal to the supremal value of the quantity {\sum_{d \in {\mathcal D}} \lambda^-_d X_d}, as {\nu^- = \sum_{d \in {\mathcal D}} \lambda^-_d 1_{E_d}} ranges over all lower bound sieves.

Proof: We prove part (i) only, and leave part (ii) as an exercise. Let {A} be the supremal value of the quantity (1) given the constraints in Problem 3, and let {B} be the infimal value of {\sum_{d \in {\mathcal D}} \lambda^+_d X_d}. We need to show that {A=B}.

We first establish the easy inequality {A \leq B}. If the sequence {a_n} obeys the constraints in Problem 3, and {\nu^+ = \sum_{d \in {\mathcal D}} \lambda^+_d 1_{E_d}} is an upper bound sieve, then

\displaystyle  \sum_n \nu^+(n) a_n = \sum_{d \in {\mathcal D}} \lambda^+_d X_d

and hence (by the non-negativity of {\nu^+} and {a_n})

\displaystyle  \sum_{n \not \in \bigcup_{p|P} E_p} a_n \leq \sum_{d \in {\mathcal D}} \lambda^+_d X_d;

taking suprema in {f} and infima in {\nu^+} we conclude that {A \leq B}.

Now suppose for contradiction that {A<B}, thus {A < C < B} for some real number {C}. We will argue using the hyperplane separation theorem; one can also proceed using one of the other duality results mentioned above. (See this previous blog post for some discussion of the connections between these various forms of linear duality.) Consider the affine functional

\displaystyle  \rho_0: (a_n)_{n \in{\bf Z}} \mapsto C - \sum_{n \not \in \bigcup_{p|P} E_p} a_n.

on the vector space of finitely supported sequences {(a_n)_{n \in {\bf Z}}} of reals. On the one hand, since {C > A}, this functional is positive for every sequence {(a_n)_{n \in{\bf Z}}} obeying the constraints in Problem 3. Next, let {K} be the space of affine functionals {\rho} of the form

\displaystyle  \rho: (a_n)_{n \in {\bf Z}} \mapsto -\sum_{d \in {\mathcal D}} \lambda^+_d ( \sum_{n \in E_d} a_n - X_d ) + \sum_n a_n \nu(n) + X

for some real numbers {\lambda^+_d \in {\bf R}}, some non-negative function {\nu: {\bf Z} \rightarrow {\bf R}^+} which is a finite linear combination of the {1_{E_d}} for {d|P}, and some non-negative {X}. This is a closed convex cone in a finite-dimensional vector space {V}; note also that {\rho_0} lies in {V}. Suppose first that {\rho_0 \in K}, thus we have a representation of the form

\displaystyle C - \sum_{n \not \in \bigcup_{p|P} E_p} a_n = -\sum_{d \in {\mathcal D}} \lambda^+_d ( \sum_{n \in E_d} a_n - X_d ) + \sum_n a_n \nu(n) + X

for any finitely supported sequence {(a_n)_{n \in {\bf Z}}}. Comparing coefficients, we conclude that

\displaystyle  \sum_{d \in {\mathcal D}} \lambda^+_d 1_{E_d}(n) \geq 1_{n \not \in \bigcup_{p|P} E_p}

for any {n} (i.e., {\sum_{d \in {\mathcal D}} \lambda^+_d 1_{E_d}} is an upper bound sieve), and also

\displaystyle  C \geq \sum_{d \in {\mathcal D}} \lambda^+_d X_d,

and thus {C \geq B}, a contradiction. Thus {\rho_0} lies outside of {K}. But then by the hyperplane separation theorem, we can find an affine functional {\iota: V \rightarrow {\bf R}} on {V} that is non-negative on {K} and negative on {\rho_0}. By duality, such an affine functional takes the form {\iota: \rho \mapsto \rho((b_n)_{n \in {\bf Z}}) + c} for some finitely supported sequence {(b_n)_{n \in {\bf Z}}} and {c \in {\bf R}} (indeed, {(b_n)_{n \in {\bf Z}}} can be supported on a finite set consisting of a single representative for each atom of the finite {\sigma}-algebra generated by the {E_p}). Since {\iota} is non-negative on the cone {K}, we see (on testing against multiples of the functionals {(a_n)_{n \in {\bf Z}} \mapsto \sum_{n \in E_d} a_n - X_d} or {(a_n)_{n \in {\bf Z}} \mapsto a_n}) that the {b_n} and {c} are non-negative, and that {\sum_{n \in E_d} b_n - X_d = 0} for all {d \in {\mathcal D}}; thus {(b_n)_{n \in {\bf Z}}} is feasible for Problem 3. Since {\iota} is negative on {\rho_0}, we see that

\displaystyle  \sum_{n \not \in \bigcup_{p|P} E_p} b_n \geq C

and thus {A \geq C}, giving the desired contradiction. \Box

Exercise 6 Prove part (ii) of the above theorem.

Exercise 7 Show that the infima and suprema in the above theorem are actually attained (so one can replace “infimal” and “supremal” by “minimal” and “maximal” if desired).

Exercise 8 What are the optimal upper and lower bound sieves for Exercise 4?

In the case when {{\mathcal D}} consists of all the divisors of {P}, we see that the Legendre sieve {\sum_{d|P} \mu(d) 1_{E_d}} is both the optimal upper bound sieve and the optimal lower bound sieve, regardless of what the quantities {X_d} are. However, in most cases of interest, {{\mathcal D}} will only be some strict subset of the divisors of {P}, and there will be a gap between the optimal upper and lower bounds.

Observe that a sequence {(\lambda^+_d)_{d \in {\mathcal D}}} of real numbers will form an upper bound sieve {\sum_d \lambda^+_d 1_{E_d}} if one has the inequalities

\displaystyle  \lambda^+_1 \geq 1


\displaystyle  \sum_{d|n} \lambda^+_d \geq 0

for all {n|P}; we will refer to such sequences as upper bound sieve coefficients. (Conversely, if the sets {E_p} are in “general position” in the sense that every set of the form {\bigcap_{p|n} E_p \backslash \bigcup_{p|P; p\not | n} E_p} for {n|P} is non-empty, we see that every upper bound sieve arises from a sequence of upper bound sieve coefficients.) Similarly, a sequence {(\lambda^-_d)_{d \in {\mathcal D}}} of real numbers will form a lower bound sieve {\sum_d \lambda^-_d 1_{E_d}} if one has the inequalities

\displaystyle  \lambda^-_1 \leq 1


\displaystyle  \sum_{d|n} \lambda^-_d \leq 0

for all {n|P} with {n>1}; we will refer to such sequences as lower bound sieve coefficients.

Exercise 9 (Brun pure sieve) Let {P} be a squarefree number, and {k} a non-negative integer. Show that the sequence {(\lambda_d)_{d \in P}} defined by

\displaystyle  \lambda_d := 1_{\omega(d) \leq k} \mu(d),

where {\omega(d)} is the number of prime factors of {d}, is a sequence of upper bound sieve coefficients for even {k}, and a sequence of lower bound sieve coefficients for odd {k}. Deduce the Bonferroni inequalities

\displaystyle  \sum_{n \in {\bf Z}} a_n 1_{n \not \in \bigcup_{p | P} E_p} \leq \sum_{d|P: \omega(p) \leq k} \mu(d) X_d \ \ \ \ \ (11)

when {k} is even, and

\displaystyle  \sum_{n \in {\bf Z}} a_n 1_{n \not \in \bigcup_{p | P} E_p} \geq \sum_{d|P: \omega(p) \leq k} \mu(d) X_d \ \ \ \ \ (12)

when {k} is odd, whenever one is in the situation of Problem 3 (and {{\mathcal D}} contains all {d|P} with {\omega(p) \leq k}). The resulting upper and lower bound sieves are sometimes known as Brun pure sieves. The Legendre sieve can be viewed as the limiting case when {k \geq \omega(P)}.

In many applications the sums {X_d} in (9) take the form

\displaystyle  \sum_{n \in E_d} a_n = g(d) X + r_d \ \ \ \ \ (13)

for some quantity {X} independent of {d}, some multiplicative function {g} with {0 \leq g(p) \leq 1}, and some remainder term {r_d} whose effect is expected to be negligible on average if {d} is restricted to be small, e.g. less than a threshold {D}; note for instance that (5) is of this form if {D \leq x^{1-\varepsilon}} for some fixed {\varepsilon>0} (note from the divisor bound, Lemma 23 of Notes 1, that {\prod_{p|d} \omega(p) \ll x^{o(1)}} if {d \ll x^{O(1)}}). We are thus led to the following idealisation of the sieving problem, in which the remainder terms {r_d} are ignored:

Problem 10 (Idealised sieving) Let {z, D \geq 1} (we refer to {z} as the sifting level and {D} as the level of distribution), let {g} be a multiplicative function with {0 \leq g(p) \leq 1}, and let {{\mathcal D} := \{ d|P(z): d \leq D \}}. How small can one make the quantity

\displaystyle  \sum_{d \in {\mathcal D}} \lambda^+_d g(d) \ \ \ \ \ (14)

for a sequence {(\lambda^+_d)_{d \in {\mathcal D}}} of upper bound sieve coefficients, and how large can one make the quantity

\displaystyle  \sum_{d \in {\mathcal D}} \lambda^-_d g(d) \ \ \ \ \ (15)

for a sequence {(\lambda^-_d)_{d \in {\mathcal D}}} of lower bound sieve coefficients?

Thus, for instance, the trivial upper bound sieve {\lambda^+_d := 1_{d=1}} and the trivial lower bound sieve {\lambda^-_d := 0} show that (14) can equal {1} and (15) can equal {0}. Of course, one hopes to do better than these trivial bounds in many situations; usually one can improve the upper bound quite substantially, but improving the lower bound is significantly more difficult, particularly when {z} is large compared with {D}.

If the remainder terms {r_d} in (13) are indeed negligible on average for {d \leq D}, then one expects the upper and lower bounds in Problem 3 to essentially be the optimal bounds in (14) and (15) respectively, multiplied by the normalisation factor {X}. Thus Problem 10 serves as a good model problem for Problem 3, in which all the arithmetic content of the original sieving problem has been abstracted into two parameters {z,D} and a multiplicative function {g}. In many applications, {g(p)} will be approximately {\kappa/p} on the average for some fixed {\kappa>0}, known as the sieve dimension; for instance, in the twin prime sieving problem discussed above, the sieve dimension is {2}. The larger one makes the level of distribution {D} compared to {z}, the more choices one has for the upper and lower bound sieves; it is thus of interest to obtain equidistribution estimates such as (13) for {d} as large as possible. When the sequence {a_d} is of arithmetic origin (for instance, if it is the von Mangoldt function {\Lambda}), then estimates such as the Bombieri-Vinogradov theorem, Theorem 17 from Notes 3, turn out to be particularly useful in this regard; in other contexts, the required equidistribution estimates might come from other sources, such as homogeneous dynamics, or the theory of expander graphs (the latter arises in the recent theory of the affine sieve, discussed in this previous blog post). However, the sieve-theoretic tools developed in this post are not particularly sensitive to how a certain level of distribution is attained, and are generally content to use sieve axioms such as (13) as “black boxes”.

In some applications one needs to modify Problem 10 in various technical ways (e.g. in altering the product {P(z)}, the set {{\mathcal D}}, or the definition of an upper or lower sieve coefficient sequence), but to simplify the exposition we will focus on the above problem without such alterations.

As the exercise below (or the heuristic (7)) suggests, the “natural” size of (14) and (15) is given by the quantity {V(z) := \prod_{p < z} (1 - g(p))} (so that the natural size for Problem 3 is {V(z) X}):

Exercise 11 Let {z,D,g} be as in Problem 10, and set {V(z) := \prod_{p \leq z} (1 - g(p))}.

  • (i) Show that the quantity (14) is always at least {V(z)} when {(\lambda^+_d)_{d \in {\mathcal D}}} is a sequence of upper bound sieve coefficients. Similarly, show that the quantity (15) is always at most {V(z)} when {(\lambda^-_d)_{d \in {\mathcal D}}} is a sequence of lower bound sieve coefficients. (Hint: compute the expected value of {\sum_{d|n} \lambda^\pm_d} when {n} is a random factor of {P(z)} chosen according to a certain probability distribution depending on {g}.)
  • (ii) Show that (14) and (15) can both attain the value of {V(z)} when {D \geq P(z)}. (Hint: translate the Legendre sieve to this setting.)

The problem of finding good sequences of upper and lower bound sieve coefficients in order to solve problems such as Problem 10 is one of the core objectives of sieve theory, and has been intensively studied. This is more of an optimisation problem rather than a genuinely number theoretic problem; however, the optimisation problem is sufficiently complicated that it has not been solved exactly or even asymptotically, except in a few special cases. (It can be reduced to a optimisation problem involving multilinear integrals of certain unknown functions of several variables, but this problem is rather difficult to analyse further; see these lecture notes of Selberg for further discussion.) But while we do not yet have a definitive solution to this problem in general, we do have a number of good general-purpose upper and lower bound sieve coefficients that give fairly good values for (14), (15), often coming within a constant factor of the idealised value {V(z)}, and which work well for sifting levels {z} as large as a small power of the level of distribution {D}. Unfortunately, we also know of an important limitation to the sieve, known as the parity problem, that prevents one from taking {z} as large as {D^{1/2}} while still obtaining non-trivial lower bounds; as a consequence, sieve theory is not able, on its own, to sift out primes for such purposes as establishing the twin prime conjecture. However, it is still possible to use these sieves, in conjunction with additional tools, to produce various types of primes or prime patterns in some cases; examples of this include the theorem of Ben Green and myself in which an upper bound sieve is used to demonstrate the existence of primes in arbitrarily long arithmetic progressions, or the more recent theorem of Zhang in which (among other things) used an upper bound sieve was used to demonstrate the existence of infinitely many pairs of primes whose difference was bounded. In such arguments, the upper bound sieve was used not so much to count the primes or prime patterns directly, but to serve instead as a sort of “container” to efficiently envelop such prime patterns; when used in such a manner, the upper bound sieves are sometimes known as enveloping sieves. If the original sequence was supported on primes, then the enveloping sieve can be viewed as a “smoothed out indicator function” that is concentrated on almost primes, which in this context refers to numbers with no small prime factors.

In a somewhat different direction, it can be possible in some cases to break the parity barrier by assuming additional equidistribution axioms on the sequence {a_n} than just (13), in particular controlling certain bilinear sums involving {a_{nm}} rather than just linear sums of the {a_n}. This approach was in particular pursued by Friedlander and Iwaniec, leading to their theorem that there are infinitely many primes of the form {n^2+m^4}.

The study of sieves is an immense topic; see for instance the recent 527-page text by Friedlander and Iwaniec. We will limit attention to two sieves which give good general-purpose results, if not necessarily the most optimal ones:

  • (i) The beta sieve (or Rosser-Iwaniec sieve), which is a modification of the classical combinatorial sieve of Brun. (A collection of sieve coefficients {\lambda_d^{\pm}} is called combinatorial if its coefficients lie in {\{-1,0,+1\}}.) The beta sieve is a family of upper and lower bound combinatorial sieves, and are particularly useful for efficiently sieving out all primes up to a parameter {z = x^{1/u}} from a set of integers of size {x}, in the regime where {u} is moderately large, leading to what is sometimes known as the fundamental lemma of sieve theory.
  • (ii) The Selberg upper bound sieve, which is a general-purpose sieve that can serve both as an upper bound sieve for classical sieving problems, as well as an enveloping sieve for sets such as the primes. (One can also convert the Selberg upper bound sieve into a lower bound sieve in a number of ways, but we will only touch upon this briefly.) A key advantage of the Selberg sieve is that, due to the “quadratic” nature of the sieve, the difficult optimisation problem in Problem 10 is replaced with a much more tractable quadratic optimisation problem, which can often be solved for exactly.

Remark 12 It is possible to compose two sieves together, for instance by using the observation that the product of two upper bound sieves is again an upper bound sieve, or that the product of an upper bound sieve and a lower bound sieve is a lower bound sieve. Such a composition of sieves is useful in some applications, for instance if one wants to apply the fundamental lemma as a “preliminary sieve” to sieve out small primes, but then use a more precise sieve like the Selberg sieve to sieve out medium primes. We will see an example of this in later notes, when we discuss the linear beta-sieve.

We will also briefly present the (arithmetic) large sieve, which gives a rather different approach to Problem 3 in the case that each {E_p} consists of some number (typically a large number) of residue classes modulo {p}, and is powered by the (analytic) large sieve inequality of the preceding section. As an application of these methods, we will utilise the Selberg upper bound sieve as an enveloping sieve to establish Zhang’s theorem on bounded gaps between primes. Finally, we give an informal discussion of the parity barrier which gives some heuristic limitations on what sieve theory is able to accomplish with regards to counting prime patters such as twin primes.

These notes are only an introduction to the vast topic of sieve theory; more detailed discussion can be found in the Friedlander-Iwaniec text, in these lecture notes of Selberg, and in many further texts.

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A fundamental and recurring problem in analytic number theory is to demonstrate the presence of cancellation in an oscillating sum, a typical example of which might be a correlation

\displaystyle  \sum_{n} f(n) \overline{g(n)} \ \ \ \ \ (1)

between two arithmetic functions {f: {\bf N} \rightarrow {\bf C}} and {g: {\bf N} \rightarrow {\bf C}}, which to avoid technicalities we will assume to be finitely supported (or that the {n} variable is localised to a finite range, such as {\{ n: n \leq x \}}). A key example to keep in mind for the purposes of this set of notes is the twisted von Mangoldt summatory function

\displaystyle  \sum_{n \leq x} \Lambda(n) \overline{\chi(n)} \ \ \ \ \ (2)

that measures the correlation between the primes and a Dirichlet character {\chi}. One can get a “trivial” bound on such sums from the triangle inequality

\displaystyle  |\sum_{n} f(n) \overline{g(n)}| \leq \sum_{n} |f(n)| |g(n)|;

for instance, from the triangle inequality and the prime number theorem we have

\displaystyle  |\sum_{n \leq x} \Lambda(n) \overline{\chi(n)}| \leq x + o(x) \ \ \ \ \ (3)

as {x \rightarrow \infty}. But the triangle inequality is insensitive to the phase oscillations of the summands, and often we expect (e.g. from the probabilistic heuristics from Supplement 4) to be able to improve upon the trivial triangle inequality bound by a substantial amount; in the best case scenario, one typically expects a “square root cancellation” that gains a factor that is roughly the square root of the number of summands. (For instance, for Dirichlet characters {\chi} of conductor {O(x^{O(1)})}, it is expected from probabilistic heuristics that the left-hand side of (3) should in fact be {O_\varepsilon(x^{1/2+\varepsilon})} for any {\varepsilon>0}.)

It has proven surprisingly difficult, however, to establish significant cancellation in many of the sums of interest in analytic number theory, particularly if the sums do not have a strong amount of algebraic structure (e.g. multiplicative structure) which allow for the deployment of specialised techniques (such as multiplicative number theory techniques). In fact, we are forced to rely (to an embarrassingly large extent) on (many variations of) a single basic tool to capture at least some cancellation, namely the Cauchy-Schwarz inequality. In fact, in many cases the classical case

\displaystyle  |\sum_n f(n) \overline{g(n)}| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}, \ \ \ \ \ (4)

considered by Cauchy, where at least one of {f, g: {\bf N} \rightarrow {\bf C}} is finitely supported, suffices for applications. Roughly speaking, the Cauchy-Schwarz inequality replaces the task of estimating a cross-correlation between two different functions {f,g}, to that of measuring self-correlations between {f} and itself, or {g} and itself, which are usually easier to compute (albeit at the cost of capturing less cancellation). Note that the Cauchy-Schwarz inequality requires almost no hypotheses on the functions {f} or {g}, making it a very widely applicable tool.

There is however some skill required to decide exactly how to deploy the Cauchy-Schwarz inequality (and in particular, how to select {f} and {g}); if applied blindly, one loses all cancellation and can even end up with a worse estimate than the trivial bound. For instance, if one tries to bound (2) directly by applying Cauchy-Schwarz with the functions {\Lambda} and {\chi}, one obtains the bound

\displaystyle  |\sum_{n \leq x} \Lambda(n) \overline{\chi(n)}| \leq (\sum_{n \leq x} \Lambda(n)^2)^{1/2} (\sum_{n \leq x} |\chi(n)|^2)^{1/2}.

The right-hand side may be bounded by {\ll x \log^{1/2} x}, but this is worse than the trivial bound (3) by a logarithmic factor. This can be “blamed” on the fact that {\Lambda} and {\chi} are concentrated on rather different sets ({\Lambda} is concentrated on primes, while {\chi} is more or less uniformly distributed amongst the natural numbers); but even if one corrects for this (e.g. by weighting Cauchy-Schwarz with some suitable “sieve weight” that is more concentrated on primes), one still does not do any better than (3). Indeed, the Cauchy-Schwarz inequality suffers from the same key weakness as the triangle inequality: it is insensitive to the phase oscillation of the factors {f, g}.

While the Cauchy-Schwarz inequality can be poor at estimating a single correlation such as (1), its power improves when considering an average (or sum, or square sum) of multiple correlations. In this set of notes, we will focus on one such situation of this type, namely that of trying to estimate a square sum

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \ \ \ \ \ (5)

that measures the correlations of a single function {f: {\bf N} \rightarrow {\bf C}} with multiple other functions {g_j: {\bf N} \rightarrow {\bf C}}. One should think of the situation in which {f} is a “complicated” function, such as the von Mangoldt function {\Lambda}, but the {g_j} are relatively “simple” functions, such as Dirichlet characters. In the case when the {g_j} are orthonormal functions, we of course have the classical Bessel inequality:

Lemma 1 (Bessel inequality) Let {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}} be finitely supported functions obeying the orthonormality relationship

\displaystyle  \sum_n g_j(n) \overline{g_{j'}(n)} = 1_{j=j'}

for all {1 \leq j,j' \leq J}. Then for any function {f: {\bf N} \rightarrow {\bf C}}, we have

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \leq (\sum_n |f(n)|^2)^{1/2}.

For sake of comparison, if one were to apply the Cauchy-Schwarz inequality (4) separately to each summand in (5), one would obtain the bound of {J^{1/2} (\sum_n |f(n)|^2)^{1/2}}, which is significantly inferior to the Bessel bound when {J} is large. Geometrically, what is going on is this: the Cauchy-Schwarz inequality (4) is only close to sharp when {f} and {g} are close to parallel in the Hilbert space {\ell^2({\bf N})}. But if {g_1,\dots,g_J} are orthonormal, then it is not possible for any other vector {f} to be simultaneously close to parallel to too many of these orthonormal vectors, and so the inner products of {f} with most of the {g_j} should be small. (See this previous blog post for more discussion of this principle.) One can view the Bessel inequality as formalising a repulsion principle: if {f} correlates too much with some of the {g_j}, then it does not have enough “energy” to have large correlation with the rest of the {g_j}.

In analytic number theory applications, it is useful to generalise the Bessel inequality to the situation in which the {g_j} are not necessarily orthonormal. This can be accomplished via the Cauchy-Schwarz inequality:

Proposition 2 (Generalised Bessel inequality) Let {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}} be finitely supported functions, and let {\nu: {\bf N} \rightarrow {\bf R}^+} be a non-negative function. Let {f: {\bf N} \rightarrow {\bf C}} be such that {f} vanishes whenever {\nu} vanishes, we have

\displaystyle  (\sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}|^2)^{1/2} \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2} \ \ \ \ \ (6)

\displaystyle  \times ( \sum_{j=1}^J \sum_{j'=1}^J c_j \overline{c_{j'}} \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)} )^{1/2}

for some sequence {c_1,\dots,c_J} of complex numbers with {\sum_{j=1}^J |c_j|^2 = 1}, with the convention that {|f(n)|^2/\nu(n)} vanishes whenever {f(n), \nu(n)} both vanish.

Note by relabeling that we may replace the domain {{\bf N}} here by any other at most countable set, such as the integers {{\bf Z}}. (Indeed, one can give an analogue of this lemma on arbitrary measure spaces, but we will not do so here.) This result first appears in this paper of Boas.

Proof: We use the method of duality to replace the role of the function {f} by a dual sequence {c_1,\dots,c_J}. By the converse to Cauchy-Schwarz, we may write the left-hand side of (6) as

\displaystyle  \sum_{j=1}^J \overline{c_j} \sum_{n} f(n) \overline{g_j(n)}

for some complex numbers {c_1,\dots,c_J} with {\sum_{j=1}^J |c_j|^2 = 1}. Indeed, if all of the {\sum_{n} f(n) \overline{g_j(n)}} vanish, we can set the {c_j} arbitrarily, otherwise we set {(c_1,\dots,c_J)} to be the unit vector formed by dividing {(\sum_{n} f(n) \overline{g_j(n)})_{j=1}^J} by its length. We can then rearrange this expression as

\displaystyle  \sum_n f(n) \overline{\sum_{j=1}^J c_j g_j(n)}.

Applying Cauchy-Schwarz (dividing the first factor by {\nu(n)^{1/2}} and multiplying the second by {\nu(n)^{1/2}}, after first removing those {n} for which {\nu(n)} vanish), this is bounded by

\displaystyle  (\sum_n |f(n)|^2 / \nu(n))^{1/2} (\sum_n \nu(n) |\sum_{j=1}^J c_j g_j(n)|^2)^{1/2},

and the claim follows by expanding out the second factor. \Box

Observe that Lemma 1 is a special case of Proposition 2 when {\nu=1} and the {g_j} are orthonormal. In general, one can expect Proposition 2 to be useful when the {g_j} are almost orthogonal relative to {\nu}, in that the correlations {\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}} tend to be small when {j,j'} are distinct. In that case, one can hope for the diagonal term {j=j'} in the right-hand side of (6) to dominate, in which case one can obtain estimates of comparable strength to the classical Bessel inequality. The flexibility to choose different weights {\nu} in the above proposition has some technical advantages; for instance, if {f} is concentrated in a sparse set (such as the primes), it is sometimes useful to tailor {\nu} to a comparable set (e.g. the almost primes) in order not to lose too much in the first factor {\sum_n |f(n)|^2 / \nu(n)}. Also, it can be useful to choose a fairly “smooth” weight {\nu}, in order to make the weighted correlations {\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}} small.

Remark 3 In harmonic analysis, the use of tools such as Proposition 2 is known as the method of almost orthogonality, or the {TT^*} method. The explanation for the latter name is as follows. For sake of exposition, suppose that {\nu} is never zero (or we remove all {n} from the domain for which {\nu(n)} vanishes). Given a family of finitely supported functions {g_1,\dots,g_J: {\bf N} \rightarrow {\bf C}}, consider the linear operator {T: \ell^2(\nu^{-1}) \rightarrow \ell^2(\{1,\dots,J\})} defined by the formula

\displaystyle  T f := ( \sum_{n} f(n) \overline{g_j(n)} )_{j=1}^J.

This is a bounded linear operator, and the left-hand side of (6) is nothing other than the {\ell^2(\{1,\dots,J\})} norm of {Tf}. Without any further information on the function {f} other than its {\ell^2(\nu^{-1})} norm {(\sum_n |f(n)|^2 / \nu(n))^{1/2}}, the best estimate one can obtain on (6) here is clearly

\displaystyle  (\sum_n |f(n)|^2 / \nu(n))^{1/2} \times \|T\|_{op},

where {\|T\|_{op}} denotes the operator norm of {T}.

The adjoint {T^*: \ell^2(\{1,\dots,J\}) \rightarrow \ell^2(\nu^{-1})} is easily computed to be

\displaystyle  T^* (c_j)_{j=1}^J := (\sum_{j=1}^J c_j \nu(n) g_j(n) )_{n \in {\bf N}}.

The composition {TT^*: \ell^2(\{1,\dots,J\}) \rightarrow \ell^2(\{1,\dots,J\})} of {T} and its adjoint is then given by

\displaystyle  TT^* (c_j)_{j=1}^J := (\sum_{j=1}^J c_j \sum_n \nu(n) g_j(n) \overline{g_{j'}}(n) )_{j=1}^J.

From the spectral theorem (or singular value decomposition), one sees that the operator norms of {T} and {TT^*} are related by the identity

\displaystyle  \|T\|_{op} = \|TT^*\|_{op}^{1/2},

and as {TT^*} is a self-adjoint, positive semi-definite operator, the operator norm {\|TT^*\|_{op}} is also the supremum of the quantity

\displaystyle  \langle TT^* (c_j)_{j=1}^J, (c_j)_{j=1}^J \rangle_{\ell^2(\{1,\dots,J\})} = \sum_{j=1}^J \sum_{j'=1}^J c_j \overline{c_{j'}} \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}

where {(c_j)_{j=1}^J} ranges over unit vectors in {\ell^2(\{1,\dots,J\})}. Putting these facts together, we obtain Proposition 2; furthermore, we see from this analysis that the bound here is essentially optimal if the only information one is allowed to use about {f} is its {\ell^2(\nu^{-1})} norm.

For further discussion of almost orthogonality methods from a harmonic analysis perspective, see Chapter VII of this text of Stein.

Exercise 4 Under the same hypotheses as Proposition 2, show that

\displaystyle  \sum_{j=1}^J |\sum_{n} f(n) \overline{g_j(n)}| \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2}

\displaystyle  \times ( \sum_{j=1}^J \sum_{j'=1}^J |\sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}| )^{1/2}

as well as the variant inequality

\displaystyle  |\sum_{j=1}^J \sum_{n} f(n) \overline{g_j(n)}| \leq (\sum_n |f(n)|^2 / \nu(n))^{1/2}

\displaystyle  \times | \sum_{j=1}^J \sum_{j'=1}^J \sum_n \nu(n) g_j(n) \overline{g_{j'}(n)}|^{1/2}.

Proposition 2 has many applications in analytic number theory; for instance, we will use it in later notes to control the large value of Dirichlet series such as the Riemann zeta function. One of the key benefits is that it largely eliminates the need to consider further correlations of the function {f} (other than its self-correlation {\sum_n |f(n)|^2 / \nu(n)} relative to {\nu^{-1}}, which is usually fairly easy to compute or estimate as {\nu} is usually chosen to be relatively simple); this is particularly useful if {f} is a function which is significantly more complicated to analyse than the functions {g_j}. Of course, the tradeoff for this is that one now has to deal with the coefficients {c_j}, which if anything are even less understood than {f}, since literally the only thing we know about these coefficients is their square sum {\sum_{j=1}^J |c_j|^2}. However, as long as there is enough almost orthogonality between the {g_j}, one can estimate the {c_j} by fairly crude estimates (e.g. triangle inequality or Cauchy-Schwarz) and still get reasonably good estimates.

In this set of notes, we will use Proposition 2 to prove some versions of the large sieve inequality, which controls a square-sum of correlations

\displaystyle  \sum_n f(n) e( -\xi_j n )

of an arbitrary finitely supported function {f: {\bf Z} \rightarrow {\bf C}} with various additive characters {n \mapsto e( \xi_j n)} (where {e(x) := e^{2\pi i x}}), or alternatively a square-sum of correlations

\displaystyle  \sum_n f(n) \overline{\chi_j(n)}

of {f} with various primitive Dirichlet characters {\chi_j}; it turns out that one can prove a (slightly sub-optimal) version of this inequality quite quickly from Proposition 2 if one first prepares the sum by inserting a smooth cutoff with well-behaved Fourier transform. The large sieve inequality has many applications (as the name suggests, it has particular utility within sieve theory). For the purposes of this set of notes, though, the main application we will need it for is the Bombieri-Vinogradov theorem, which in a very rough sense gives a prime number theorem in arithmetic progressions, which, “on average”, is of strength comparable to the results provided by the Generalised Riemann Hypothesis (GRH), but has the great advantage of being unconditional (it does not require any unproven hypotheses such as GRH); it can be viewed as a significant extension of the Siegel-Walfisz theorem from Notes 2. As we shall see in later notes, the Bombieri-Vinogradov theorem is a very useful ingredient in sieve-theoretic problems involving the primes.

There is however one additional important trick, beyond the large sieve, which we will need in order to establish the Bombieri-Vinogradov theorem. As it turns out, after some basic manipulations (and the deployment of some multiplicative number theory, and specifically the Siegel-Walfisz theorem), the task of proving the Bombieri-Vinogradov theorem is reduced to that of getting a good estimate on sums that are roughly of the form

\displaystyle  \sum_{j=1}^J |\sum_n \Lambda(n) \overline{\chi_j}(n)| \ \ \ \ \ (7)

for some primitive Dirichlet characters {\chi_j}. This looks like the type of sum that can be controlled by the large sieve (or by Proposition 2), except that this is an ordinary sum rather than a square sum (i.e., an {\ell^1} norm rather than an {\ell^2} norm). One could of course try to control such a sum in terms of the associated square-sum through the Cauchy-Schwarz inequality, but this turns out to be very wasteful (it loses a factor of about {J^{1/2}}). Instead, one should try to exploit the special structure of the von Mangoldt function {\Lambda}, in particular the fact that it can be expressible as a Dirichlet convolution {\alpha * \beta} of two further arithmetic sequences {\alpha,\beta} (or as a finite linear combination of such Dirichlet convolutions). The reason for introducing this convolution structure is through the basic identity

\displaystyle  (\sum_n \alpha*\beta(n) \overline{\chi_j}(n)) = (\sum_n \alpha(n) \overline{\chi_j}(n)) (\sum_n \beta(n) \overline{\chi_j}(n)) \ \ \ \ \ (8)

for any finitely supported sequences {\alpha,\beta: {\bf N} \rightarrow {\bf C}}, as can be easily seen by multiplying everything out and using the completely multiplicative nature of {\chi_j}. (This is the multiplicative analogue of the well-known relationship {\widehat{f*g}(\xi) = \hat f(\xi) \hat g(\xi)} between ordinary convolution and Fourier coefficients.) This factorisation, together with yet another application of the Cauchy-Schwarz inequality, lets one control (7) by square-sums of the sort that can be handled by the large sieve inequality.

As we have seen in Notes 1, the von Mangoldt function {\Lambda} does indeed admit several factorisations into Dirichlet convolution type, such as the factorisation {\Lambda = \mu * L}. One can try directly inserting this factorisation into the above strategy; it almost works, however there turns out to be a problem when considering the contribution of the portion of {\mu} or {L} that is supported at very small natural numbers, as the large sieve loses any gain over the trivial bound in such settings. Because of this, there is a need for a more sophisticated decomposition of {\Lambda} into Dirichlet convolutions {\alpha * \beta} which are non-degenerate in the sense that {\alpha,\beta} are supported away from small values. (As a non-example, the trivial factorisation {\Lambda = \Lambda * \delta} would be a totally inappropriate factorisation for this purpose.) Fortunately, it turns out that through some elementary combinatorial manipulations, some satisfactory decompositions of this type are available, such as the Vaughan identity and the Heath-Brown identity. By using one of these identities we will be able to complete the proof of the Bombieri-Vinogradov theorem. (These identities are also useful for other applications in which one wishes to control correlations between the von Mangoldt function {\Lambda} and some other sequence; we will see some examples of this in later notes.)

For further reading on these topics, including a significantly larger number of examples of the large sieve inequality, see Chapters 7 and 17 of Iwaniec and Kowalski.

Remark 5 We caution that the presentation given in this set of notes is highly ahistorical; we are using modern streamlined proofs of results that were first obtained by more complicated arguments.

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We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins {n,n+2}. This creates a major jump in difficulty; for instance, even the most basic multiplicative result about the primes, namely Euclid’s theorem that there are infinitely many of them, remains unproven for twin primes. Of course, the situation is even worse for stronger results, such as Euler’s theorem, Dirichlet’s theorem, or the prime number theorem. Finally, even many multiplicative questions about the primes remain open. The most famous of these is the Riemann hypothesis, which gives the asymptotic {\sum_{n \leq x} \Lambda(n) = x + O( \sqrt{x} \log^2 x )} (see Proposition 24 from Notes 2). But even if one assumes the Riemann hypothesis, the precise distribution of the error term {O( \sqrt{x} \log^2 x )} in the above asymptotic (or in related asymptotics, such as for the sum {\sum_{x \leq n < x+y} \Lambda(n)} that measures the distribution of primes in short intervals) is not entirely clear.

Despite this, we do have a number of extremely convincing and well supported models for the primes (and related objects) that let us predict what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes. These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected {k}-tuple), and replacing that distribution by a model distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models. In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!

As it turns out, the models for primes that have turned out to be the most accurate in practice are random models, which involve (either explicitly or implicitly) one or more random variables. This is despite the prime numbers being obviously deterministic in nature; no coins are flipped or dice rolled to create the set of primes. The point is that while the primes have a lot of obvious multiplicative structure (for instance, the product of two primes is never another prime), they do not appear to exhibit much discernible non-multiplicative structure asymptotically, in the sense that they rarely exhibit statistical anomalies in the asymptotic limit that cannot be easily explained in terms of the multiplicative properties of the primes. As such, when considering non-multiplicative statistics of the primes, the primes appear to behave pseudorandomly, and can thus be modeled with reasonable accuracy by a random model. And even for multiplicative problems, which are in principle controlled by the zeroes of the Riemann zeta function, one can obtain good predictions by positing various pseudorandomness properties of these zeroes, so that the distribution of these zeroes can be modeled by a random model.

Of course, one cannot expect perfect accuracy when replicating a deterministic set such as the primes by a probabilistic model of that set, and each of the heuristic models we discuss below have some limitations to the range of statistics about the primes that they can expect to track with reasonable accuracy. For instance, many of the models about the primes do not fully take into account the multiplicative structure of primes, such as the connection with a zeta function with a meromorphic continuation to the entire complex plane; at the opposite extreme, we have the GUE hypothesis which appears to accurately model the zeta function, but does not capture such basic properties of the primes as the fact that the primes are all natural numbers. Nevertheless, each of the models described below, when deployed within their sphere of reasonable application, has (possibly after some fine-tuning) given predictions that are in remarkable agreement with numerical computation and with known rigorous theoretical results, as well as with other models in overlapping spheres of application; they are also broadly compatible with the general heuristic (discussed in this previous post) that in the absence of any exploitable structure, asymptotic statistics should default to the most “uniform”, “pseudorandom”, or “independent” distribution allowable.

As hinted at above, we do not have a single unified model for the prime numbers (other than the primes themselves, of course), but instead have an overlapping family of useful models that each appear to accurately describe some, but not all, aspects of the prime numbers. In this set of notes, we will discuss four such models:

  1. The Cramér random model and its refinements, which model the set {{\mathcal P}} of prime numbers by a random set.
  2. The Möbius pseudorandomness principle, which predicts that the Möbius function {\mu} does not correlate with any genuinely different arithmetic sequence of reasonable “complexity”.
  3. The equidistribution of residues principle, which predicts that the residue classes of a large number {n} modulo a small or medium-sized prime {p} behave as if they are independently and uniformly distributed as {p} varies.
  4. The GUE hypothesis, which asserts that the zeroes of the Riemann zeta function are distributed (at microscopic and mesoscopic scales) like the zeroes of a GUE random matrix, and which generalises the pair correlation conjecture regarding pairs of such zeroes.

This is not an exhaustive list of models for the primes and related objects; for instance, there is also the model in which the major arc contribution in the Hardy-Littlewood circle method is predicted to always dominate, and with regards to various finite groups of number-theoretic importance, such as the class groups discussed in Supplement 1, there are also heuristics of Cohen-Lenstra type. Historically, the first heuristic discussion of the primes along these lines was by Sylvester, who worked informally with a model somewhat related to the equidistribution of residues principle. However, we will not discuss any of these models here.

A word of warning: the discussion of the above four models will inevitably be largely informal, and “fuzzy” in nature. While one can certainly make precise formalisations of at least some aspects of these models, one should not be inflexibly wedded to a specific such formalisation as being “the” correct way to pin down the model rigorously. (To quote the statistician George Box: “all models are wrong, but some are useful”.) Indeed, we will see some examples below the fold in which some finer structure in the prime numbers leads to a correction term being added to a “naive” implementation of one of the above models to make it more accurate, and it is perfectly conceivable that some further such fine-tuning will be applied to one or more of these models in the future. These sorts of mathematical models are in some ways closer in nature to the scientific theories used to model the physical world, than they are to the axiomatic theories one is accustomed to in rigorous mathematics, and one should approach the discussion below accordingly. In particular, and in contrast to the other notes in this course, the material here is not directly used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used indirectly for such purposes, for instance by

  • giving a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures;
  • providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts, or an error term is too small);
  • gauging the relative strength of various assertions (e.g. classifying some results as “unsurprising”, others as “potential breakthroughs” or “powerful new estimates”, others as “unexpected new phenomena”, and yet others as “way too good to be true”); or
  • setting up heuristic barriers (such as the parity barrier) that one has to resolve before resolving certain key problems (e.g. the twin prime conjecture).

See also my previous essay on the distinction between “rigorous” and “post-rigorous” mathematics, or Thurston’s essay discussing, among other things, the “definition-theorem-proof” model of mathematics and its limitations.

Remark 1 The material in this set of notes presumes some prior exposure to probability theory. See for instance this previous post for a quick review of the relevant concepts.

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In Notes 2, the Riemann zeta function {\zeta} (and more generally, the Dirichlet {L}-functions {L(\cdot,\chi)}) were extended meromorphically into the region {\{ s: \hbox{Re}(s) > 0 \}} in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are known as the non-trivial zeroes of {\zeta}, and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The {\zeta} function obeys the trivial functional equation

\displaystyle  \zeta(\overline{s}) = \overline{\zeta(s)} \ \ \ \ \ (1)

for all {s} in its domain of definition. Indeed, as {\zeta(s)} is real-valued when {s} is real, the function {\zeta(s) - \overline{\zeta(\overline{s})}} vanishes on the real line and is also meromorphic, and hence vanishes everywhere. Similarly one has the functional equation

\displaystyle  \overline{L(s, \chi)} = L(\overline{s}, \overline{\chi}). \ \ \ \ \ (2)

From these equations we see that the zeroes of the zeta function are symmetric across the real axis, and the zeroes of {L(\cdot,\chi)} are the reflection of the zeroes of {L(\cdot,\overline{\chi})} across this axis.

It is a remarkable fact that these functions obey an additional, and more non-trivial, functional equation, this time establishing a symmetry across the critical line {\{ s: \hbox{Re}(s) = \frac{1}{2} \}} rather than the real axis. One consequence of this symmetry is that the zeta function and {L}-functions may be extended meromorphically to the entire complex plane. For the zeta function, the functional equation was discovered by Riemann, and reads as follows:

Theorem 1 (Functional equation for the Riemann zeta function) The Riemann zeta function {\zeta} extends meromorphically to the entire complex plane, with a simple pole at {s=1} and no other poles. Furthermore, one has the functional equation

\displaystyle  \zeta(s) = \alpha(s) \zeta(1-s) \ \ \ \ \ (3)

or equivalently

\displaystyle  \zeta(1-s) = \alpha(1-s) \zeta(s) \ \ \ \ \ (4)

for all complex {s} other than {s=0,1}, where {\alpha} is the function

\displaystyle  \alpha(s) := 2^s \pi^{s-1} \sin( \frac{\pi s}{2}) \Gamma(1-s). \ \ \ \ \ (5)

Here {\cos(z) := \frac{e^z + e^{-z}}{2}}, {\sin(z) := \frac{e^{-z}-e^{-z}}{2i}} are the complex-analytic extensions of the classical trigionometric functions {\cos(x), \sin(x)}, and {\Gamma} is the Gamma function, whose definition and properties we review below the fold.

The functional equation can be placed in a more symmetric form as follows:

Corollary 2 (Functional equation for the Riemann xi function) The Riemann xi function

\displaystyle  \xi(s) := \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s) \ \ \ \ \ (6)

is analytic on the entire complex plane {{\bf C}} (after removing all removable singularities), and obeys the functional equations

\displaystyle  \xi(\overline{s}) = \overline{\xi(s)}


\displaystyle  \xi(s) = \xi(1-s). \ \ \ \ \ (7)

In particular, the zeroes of {\xi} consist precisely of the non-trivial zeroes of {\zeta}, and are symmetric about both the real axis and the critical line. Also, {\xi} is real-valued on the critical line and on the real axis.

Corollary 2 is an easy consequence of Theorem 1 together with the duplication theorem for the Gamma function, and the fact that {\zeta} has no zeroes to the right of the critical strip, and is left as an exercise to the reader (Exercise 19). The functional equation in Theorem 1 has many proofs, but most of them are related in on way or another to the Poisson summation formula

\displaystyle  \sum_n f(n) = \sum_m \hat f(2\pi m) \ \ \ \ \ (8)

(Theorem 34 from Supplement 2, at least in the case when {f} is twice continuously differentiable and compactly supported), which can be viewed as a Fourier-analytic link between the coarse-scale distribution of the integers and the fine-scale distribution of the integers. Indeed, there is a quick heuristic proof of the functional equation that comes from formally applying the Poisson summation formula to the function {1_{x>0} \frac{1}{x^s}}, and noting that the functions {x \mapsto \frac{1}{x^s}} and {\xi \mapsto \frac{1}{\xi^{1-s}}} are formally Fourier transforms of each other, up to some Gamma function factors, as well as some trigonometric factors arising from the distinction between the real line and the half-line. Such a heuristic proof can indeed be made rigorous, and we do so below the fold, while also providing Riemann’s two classical proofs of the functional equation.

From the functional equation (and the poles of the Gamma function), one can see that {\zeta} has trivial zeroes at the negative even integers {-2,-4,-6,\dots}, in addition to the non-trivial zeroes in the critical strip. More generally, the following table summarises the zeroes and poles of the various special functions appearing in the functional equation, after they have been meromorphically extended to the entire complex plane, and with zeroes classified as “non-trivial” or “trivial” depending on whether they lie in the critical strip or not. (Exponential functions such as {2^{s-1}} or {\pi^{-s}} have no zeroes or poles, and will be ignored in this table; the zeroes and poles of rational functions such as {s(s-1)} are self-evident and will also not be displayed here.)

Function Non-trivial zeroes Trivial zeroes Poles
{\zeta(s)} Yes {-2,-4,-6,\dots} {1}
{\zeta(1-s)} Yes {1,3,5,\dots} {0}
{\sin(\pi s/2)} No Even integers No
{\cos(\pi s/2)} No Odd integers No
{\sin(\pi s)} No Integers No
{\Gamma(s)} No No {0,-1,-2,\dots}
{\Gamma(s/2)} No No {0,-2,-4,\dots}
{\Gamma(1-s)} No No {1,2,3,\dots}
{\Gamma((1-s)/2)} No No {2,4,6,\dots}
{\xi(s)} Yes No No

Among other things, this table indicates that the Gamma and trigonometric factors in the functional equation are tied to the trivial zeroes and poles of zeta, but have no direct bearing on the distribution of the non-trivial zeroes, which is the most important feature of the zeta function for the purposes of analytic number theory, beyond the fact that they are symmetric about the real axis and critical line. In particular, the Riemann hypothesis is not going to be resolved just from further analysis of the Gamma function!

The zeta function computes the “global” sum {\sum_n \frac{1}{n^s}}, with {n} ranging all the way from {1} to infinity. However, by some Fourier-analytic (or complex-analytic) manipulation, it is possible to use the zeta function to also control more “localised” sums, such as {\sum_n \frac{1}{n^s} \psi(\log n - \log N)} for some {N \gg 1} and some smooth compactly supported function {\psi: {\bf R} \rightarrow {\bf C}}. It turns out that the functional equation (3) for the zeta function localises to this context, giving an approximate functional equation which roughly speaking takes the form

\displaystyle  \sum_n \frac{1}{n^s} \psi( \log n - \log N ) \approx \alpha(s) \sum_m \frac{1}{m^{1-s}} \psi( \log M - \log m )

whenever {s=\sigma+it} and {NM = \frac{|t|}{2\pi}}; see Theorem 38 below for a precise formulation of this equation. Unsurprisingly, this form of the functional equation is also very closely related to the Poisson summation formula (8), indeed it is essentially a special case of that formula (or more precisely, of the van der Corput {B}-process). This useful identity relates long smoothed sums of {\frac{1}{n^s}} to short smoothed sums of {\frac{1}{m^{1-s}}} (or vice versa), and can thus be used to shorten exponential sums involving terms such as {\frac{1}{n^s}}, which is useful when obtaining some of the more advanced estimates on the Riemann zeta function.

We will give two other basic uses of the functional equation. The first is to get a good count (as opposed to merely an upper bound) on the density of zeroes in the critical strip, establishing the Riemann-von Mangoldt formula that the number {N(T)} of zeroes of imaginary part between {0} and {T} is {\frac{T}{2\pi} \log \frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T)} for large {T}. The other is to obtain untruncated versions of the explicit formula from Notes 2, giving a remarkable exact formula for sums involving the von Mangoldt function in terms of zeroes of the Riemann zeta function. These results are not strictly necessary for most of the material in the rest of the course, but certainly help to clarify the nature of the Riemann zeta function and its relation to the primes.

In view of the material in previous notes, it should not be surprising that there are analogues of all of the above theory for Dirichlet {L}-functions {L(\cdot,\chi)}. We will restrict attention to primitive characters {\chi}, since the {L}-function for imprimitive characters merely differs from the {L}-function of the associated primitive factor by a finite Euler product; indeed, if {\chi = \chi' \chi_0} for some principal {\chi_0} whose modulus {q_0} is coprime to that of {\chi'}, then

\displaystyle  L(s,\chi) = L(s,\chi') \prod_{p|q_0} (1 - \frac{1}{p^s}) \ \ \ \ \ (9)

(cf. equation (45) of Notes 2).

The main new feature is that the Poisson summation formula needs to be “twisted” by a Dirichlet character {\chi}, and this boils down to the problem of understanding the finite (additive) Fourier transform of a Dirichlet character. This is achieved by the classical theory of Gauss sums, which we review below the fold. There is one new wrinkle; the value of {\chi(-1) \in \{-1,+1\}} plays a role in the functional equation. More precisely, we have

Theorem 3 (Functional equation for {L}-functions) Let {\chi} be a primitive character of modulus {q} with {q>1}. Then {L(s,\chi)} extends to an entire function on the complex plane, with

\displaystyle  L(s,\chi) = \varepsilon(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin(\frac{\pi}{2}(s+\kappa)) \Gamma(1-s) L(1-s,\overline{\chi})

or equivalently

\displaystyle  L(1-s,\overline{\chi}) = \varepsilon(\overline{\chi}) 2^{1-s} \pi^{-s} q^{s-1/2} \sin(\frac{\pi}{2}(1-s+\kappa)) \Gamma(s) L(s,\chi)

for all {s}, where {\kappa} is equal to {0} in the even case {\chi(-1)=+1} and {1} in the odd case {\chi(-1)=-1}, and

\displaystyle  \varepsilon(\chi) := \frac{\tau(\chi)}{i^\kappa \sqrt{q}} \ \ \ \ \ (10)

where {\tau(\chi)} is the Gauss sum

\displaystyle  \tau(\chi) := \sum_{n \in {\bf Z}/q{\bf Z}} \chi(n) e(n/q). \ \ \ \ \ (11)

and {e(x) := e^{2\pi ix}}, with the convention that the {q}-periodic function {n \mapsto e(n/q)} is also (by abuse of notation) applied to {n} in the cyclic group {{\bf Z}/q{\bf Z}}.

From this functional equation and (2) we see that, as with the Riemann zeta function, the non-trivial zeroes of {L(s,\chi)} (defined as the zeroes within the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are symmetric around the critical line (and, if {\chi} is real, are also symmetric around the real axis). In addition, {L(s,\chi)} acquires trivial zeroes at the negative even integers and at zero if {\chi(-1)=1}, and at the negative odd integers if {\chi(-1)=-1}. For imprimitive {\chi}, we see from (9) that {L(s,\chi)} also acquires some additional trivial zeroes on the left edge of the critical strip.

There is also a symmetric version of this equation, analogous to Corollary 2:

Corollary 4 Let {\chi,q,\varepsilon(\chi)} be as above, and set

\displaystyle  \xi(s,\chi) := (q/\pi)^{(s+\kappa)/2} \Gamma((s+\kappa)/2) L(s,\chi),

then {\xi(\cdot,\chi)} is entire with {\xi(1-s,\chi) = \varepsilon(\chi) \xi(s,\chi)}.

For further detail on the functional equation and its implications, I recommend the classic text of Titchmarsh or the text of Davenport.

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In Notes 1, we approached multiplicative number theory (the study of multiplicative functions {f: {\bf N} \rightarrow {\bf C}} and their relatives) via elementary methods, in which attention was primarily focused on obtaining asymptotic control on summatory functions {\sum_{n \leq x} f(n)} and logarithmic sums {\sum_{n \leq x} \frac{f(n)}{n}}. Now we turn to the complex approach to multiplicative number theory, in which the focus is instead on obtaining various types of control on the Dirichlet series {{\mathcal D} f}, defined (at least for {s} of sufficiently large real part) by the formula

\displaystyle  {\mathcal D} f(s) := \sum_n \frac{f(n)}{n^s}.

These series also made an appearance in the elementary approach to the subject, but only for real {s} that were larger than {1}. But now we will exploit the freedom to extend the variable {s} to the complex domain; this gives enough freedom (in principle, at least) to recover control of elementary sums such as {\sum_{n\leq x} f(n)} or {\sum_{n\leq x} \frac{f(n)}{n}} from control on the Dirichlet series. Crucially, for many key functions {f} of number-theoretic interest, the Dirichlet series {{\mathcal D} f} can be analytically (or at least meromorphically) continued to the left of the line {\{ s: \hbox{Re}(s) = 1 \}}. The zeroes and poles of the resulting meromorphic continuations of {{\mathcal D} f} (and of related functions) then turn out to control the asymptotic behaviour of the elementary sums of {f}; the more one knows about the former, the more one knows about the latter. In particular, knowledge of where the zeroes of the Riemann zeta function {\zeta} are located can give very precise information about the distribution of the primes, by means of a fundamental relationship known as the explicit formula. There are many ways of phrasing this explicit formula (both in exact and in approximate forms), but they are all trying to formalise an approximation to the von Mangoldt function {\Lambda} (and hence to the primes) of the form

\displaystyle  \Lambda(n) \approx 1 - \sum_\rho n^{\rho-1} \ \ \ \ \ (1)

where the sum is over zeroes {\rho} (counting multiplicity) of the Riemann zeta function {\zeta = {\mathcal D} 1} (with the sum often restricted so that {\rho} has large real part and bounded imaginary part), and the approximation is in a suitable weak sense, so that

\displaystyle  \sum_n \Lambda(n) g(n) \approx \int_0^\infty g(y)\ dy - \sum_\rho \int_0^\infty g(y) y^{\rho-1}\ dy \ \ \ \ \ (2)

for suitable “test functions” {g} (which in practice are restricted to be fairly smooth and slowly varying, with the precise amount of restriction dependent on the amount of truncation in the sum over zeroes one wishes to take). Among other things, such approximations can be used to rigorously establish the prime number theorem

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + o(x) \ \ \ \ \ (3)

as {x \rightarrow \infty}, with the size of the error term {o(x)} closely tied to the location of the zeroes {\rho} of the Riemann zeta function.

The explicit formula (1) (or any of its more rigorous forms) is closely tied to the counterpart approximation

\displaystyle  -\frac{\zeta'}{\zeta}(s) \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho} \ \ \ \ \ (4)

for the Dirichlet series {{\mathcal D} \Lambda = -\frac{\zeta'}{\zeta}} of the von Mangoldt function; note that (4) is formally the special case of (2) when {g(n) = n^{-s}}. Such approximations come from the general theory of local factorisations of meromorphic functions, as discussed in Supplement 2; the passage from (4) to (2) is accomplished by such tools as the residue theorem and the Fourier inversion formula, which were also covered in Supplement 2. The relative ease of uncovering the Fourier-like duality between primes and zeroes (sometimes referred to poetically as the “music of the primes”) is one of the major advantages of the complex-analytic approach to multiplicative number theory; this important duality tends to be rather obscured in the other approaches to the subject, although it can still in principle be discernible with sufficient effort.

More generally, one has an explicit formula

\displaystyle  \Lambda(n) \chi(n) \approx - \sum_\rho n^{\rho-1} \ \ \ \ \ (5)

for any (non-principal) Dirichlet character {\chi}, where {\rho} now ranges over the zeroes of the associated Dirichlet {L}-function {L(s,\chi) := {\mathcal D} \chi(s)}; we view this formula as a “twist” of (1) by the Dirichlet character {\chi}. The explicit formula (5), proven similarly (in any of its rigorous forms) to (1), is important in establishing the prime number theorem in arithmetic progressions, which asserts that

\displaystyle  \sum_{n \leq x: n = a\ (q)} \Lambda(n) = \frac{x}{\phi(q)} + o(x) \ \ \ \ \ (6)

as {x \rightarrow \infty}, whenever {a\ (q)} is a fixed primitive residue class. Again, the size of the error term {o(x)} here is closely tied to the location of the zeroes of the Dirichlet {L}-function, with particular importance given to whether there is a zero very close to {s=1} (such a zero is known as an exceptional zero or Siegel zero).

While any information on the behaviour of zeta functions or {L}-functions is in principle welcome for the purposes of analytic number theory, some regions of the complex plane are more important than others in this regard, due to the differing weights assigned to each zero in the explicit formula. Roughly speaking, in descending order of importance, the most crucial regions on which knowledge of these functions is useful are

  1. The region on or near the point {s=1}.
  2. The region on or near the right edge {\{ 1+it: t \in {\bf R} \}} of the critical strip {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}.
  3. The right half {\{ s: \frac{1}{2} < \hbox{Re}(s) < 1 \}} of the critical strip.
  4. The region on or near the critical line {\{ \frac{1}{2} + it: t \in {\bf R} \}} that bisects the critical strip.
  5. Everywhere else.

For instance:

  1. We will shortly show that the Riemann zeta function {\zeta} has a simple pole at {s=1} with residue {1}, which is already sufficient to recover much of the classical theorems of Mertens discussed in the previous set of notes, as well as results on mean values of multiplicative functions such as the divisor function {\tau}. For Dirichlet {L}-functions, the behaviour is instead controlled by the quantity {L(1,\chi)} discussed in Notes 1, which is in turn closely tied to the existence and location of a Siegel zero.
  2. The zeta function is also known to have no zeroes on the right edge {\{1+it: t \in {\bf R}\}} of the critical strip, which is sufficient to prove (and is in fact equivalent to) the prime number theorem. Any enlargement of the zero-free region for {\zeta} into the critical strip leads to improved error terms in that theorem, with larger zero-free regions leading to stronger error estimates. Similarly for {L}-functions and the prime number theorem in arithmetic progressions.
  3. The (as yet unproven) Riemann hypothesis prohibits {\zeta} from having any zeroes within the right half {\{ s: \frac{1}{2} < \hbox{Re}(s) < 1 \}} of the critical strip, and gives very good control on the number of primes in intervals, even when the intervals are relatively short compared to the size of the entries. Even without assuming the Riemann hypothesis, zero density estimates in this region are available that give some partial control of this form. Similarly for {L}-functions, primes in short arithmetic progressions, and the generalised Riemann hypothesis.
  4. Assuming the Riemann hypothesis, further distributional information about the zeroes on the critical line (such as Montgomery’s pair correlation conjecture, or the more general GUE hypothesis) can give finer information about the error terms in the prime number theorem in short intervals, as well as other arithmetic information. Again, one has analogues for {L}-functions and primes in short arithmetic progressions.
  5. The functional equation of the zeta function describes the behaviour of {\zeta} to the left of the critical line, in terms of the behaviour to the right of the critical line. This is useful for building a “global” picture of the structure of the zeta function, and for improving a number of estimates about that function, but (in the absence of unproven conjectures such as the Riemann hypothesis or the pair correlation conjecture) it turns out that many of the basic analytic number theory results using the zeta function can be established without relying on this equation. Similarly for {L}-functions.

Remark 1 If one takes an “adelic” viewpoint, one can unite the Riemann zeta function {\zeta(\sigma+it) = \sum_n n^{-\sigma-it}} and all of the {L}-functions {L(\sigma+it,\chi) = \sum_n \chi(n) n^{-\sigma-it}} for various Dirichlet characters {\chi} into a single object, viewing {n \mapsto \chi(n) n^{-it}} as a general multiplicative character on the adeles; thus the imaginary coordinate {t} and the Dirichlet character {\chi} are really the Archimedean and non-Archimedean components respectively of a single adelic frequency parameter. This viewpoint was famously developed in Tate’s thesis, which among other things helps to clarify the nature of the functional equation, as discussed in this previous post. We will not pursue the adelic viewpoint further in these notes, but it does supply a “high-level” explanation for why so much of the theory of the Riemann zeta function extends to the Dirichlet {L}-functions. (The non-Archimedean character {\chi(n)} and the Archimedean character {n^{it}} behave similarly from an algebraic point of view, but not so much from an analytic point of view; as such, the adelic viewpoint is well suited for algebraic tasks (such as establishing the functional equation), but not for analytic tasks (such as establishing a zero-free region).)

Roughly speaking, the elementary multiplicative number theory from Notes 1 corresponds to the information one can extract from the complex-analytic method in region 1 of the above hierarchy, while the more advanced elementary number theory used to prove the prime number theorem (and which we will not cover in full detail in these notes) corresponds to what one can extract from regions 1 and 2.

As a consequence of this hierarchy of importance, information about the {\zeta} function away from the critical strip, such as Euler’s identity

\displaystyle  \zeta(2) = \frac{\pi^2}{6}

or equivalently

\displaystyle  1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots = \frac{\pi^2}{6}

or the infamous identity

\displaystyle  \zeta(-1) = -\frac{1}{12},

which is often presented (slightly misleadingly, if one’s conventions for divergent summation are not made explicit) as

\displaystyle  1 + 2 + 3 + \dots = -\frac{1}{12},

are of relatively little direct importance in analytic prime number theory, although they are still of interest for some other, non-number-theoretic, applications. (The quantity {\zeta(2)} does play a minor role as a normalising factor in some asymptotics, see e.g. Exercise 28 from Notes 1, but its precise value is usually not of major importance.) In contrast, the value {L(1,\chi)} of an {L}-function at {s=1} turns out to be extremely important in analytic number theory, with many results in this subject relying ultimately on a non-trivial lower-bound on this quantity coming from Siegel’s theorem, discussed below the fold.

For a more in-depth treatment of the topics in this set of notes, see Davenport’s “Multiplicative number theory“.

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We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. We will not attempt a comprehensive review of this subject; for instance, we will completely neglect the conformal geometry or Riemann surface aspect of complex analysis, and we will also avoid using the various boundary convergence theorems for Taylor series or Dirichlet series (the latter type of result is traditionally utilised in multiplicative number theory, but I personally find them a little unintuitive to use, and will instead rely on a slightly different set of complex-analytic tools). We will also focus on the “local” structure of complex analytic functions, in particular adopting the philosophy that such functions behave locally like complex polynomials; the classical “global” theory of entire functions, while traditionally used in the theory of the Riemann zeta function, will be downplayed in these notes. On the other hand, we will play up the relationship between complex analysis and Fourier analysis, as we will incline to using the latter tool over the former in some of the subsequent material. (In the traditional approach to the subject, the Mellin transform is used in place of the Fourier transform, but we will not emphasise the role of the Mellin transform here.)

We begin by recalling the notion of a holomorphic function, which will later be shown to be essentially synonymous with that of a complex analytic function.

Definition 1 (Holomorphic function) Let {\Omega} be an open subset of {{\bf C}}, and let {f: \Omega \rightarrow {\bf C}} be a function. If {z \in {\bf C}}, we say that {f} is complex differentiable at {z} if the limit

\displaystyle  f'(z) := \lim_{h \rightarrow 0; h \in {\bf C} \backslash \{0\}} \frac{f(z+h)-f(z)}{h}

exists, in which case we refer to {f'(z)} as the (complex) derivative of {f} at {z}. If {f} is differentiable at every point {z} of {\Omega}, and the derivative {f': \Omega \rightarrow {\bf C}} is continuous, we say that {f} is holomorphic on {\Omega}.

Exercise 2 Show that a function {f: \Omega \rightarrow {\bf C}} is holomorphic if and only if the two-variable function {(x,y) \mapsto f(x+iy)} is continuously differentiable on {\{ (x,y) \in {\bf R}^2: x+iy \in \Omega\}} and obeys the Cauchy-Riemann equation

\displaystyle  \frac{\partial}{\partial x} f(x+iy) = \frac{1}{i} \frac{\partial}{\partial y} f(x+iy). \ \ \ \ \ (1)

Basic examples of holomorphic functions include complex polynomials

\displaystyle  P(z) = a_n z^n + \dots + a_1 z + a_0

as well as the complex exponential function

\displaystyle  \exp(z) := \sum_{n=0}^\infty \frac{z^n}{n!}

which are holomorphic on the entire complex plane {{\bf C}} (i.e., they are entire functions). The sum or product of two holomorphic functions is again holomorphic; the quotient of two holomorphic functions is holomorphic so long as the denominator is non-zero. Finally, the composition of two holomorphic functions is holomorphic wherever the composition is defined.

Exercise 3

  • (i) Establish Euler’s formula

    \displaystyle  \exp(x+iy) = e^x (\cos y + i \sin y)

    for all {x,y \in {\bf R}}. (Hint: it is a bit tricky to do this starting from the trigonometric definitions of sine and cosine; I recommend either using the Taylor series formulations of these functions instead, or alternatively relying on the ordinary differential equations obeyed by sine and cosine.)

  • (ii) Show that every non-zero complex number {z} has a complex logarithm {\log(z)} such that {\exp(\log(z))=z}, and that this logarithm is unique up to integer multiples of {2\pi i}.
  • (iii) Show that there exists a unique principal branch {\hbox{Log}(z)} of the complex logarithm in the region {{\bf C} \backslash (-\infty,0]}, defined by requiring {\hbox{Log}(z)} to be a logarithm of {z} with imaginary part between {-\pi} and {\pi}. Show that this principal branch is holomorphic with derivative {1/z}.

In real analysis, we have the fundamental theorem of calculus, which asserts that

\displaystyle  \int_a^b F'(t)\ dt = F(b) - F(a)

whenever {[a,b]} is a real interval and {F: [a,b] \rightarrow {\bf R}} is a continuously differentiable function. The complex analogue of this fact is that

\displaystyle  \int_\gamma F'(z)\ dz = F(\gamma(1)) - F(\gamma(0)) \ \ \ \ \ (2)

whenever {F: \Omega \rightarrow {\bf C}} is a holomorphic function, and {\gamma: [0,1] \rightarrow \Omega} is a contour in {\Omega}, by which we mean a piecewise continuously differentiable function, and the contour integral {\int_\gamma f(z)\ dz} for a continuous function {f} is defined via change of variables as

\displaystyle  \int_\gamma f(z)\ dz := \int_0^1 f(\gamma(t)) \gamma'(t)\ dt.

The complex fundamental theorem of calculus (2) follows easily from the real fundamental theorem and the chain rule.

In real analysis, we have the rather trivial fact that the integral of a continuous function on a closed contour is always zero:

\displaystyle  \int_a^b f(t)\ dt + \int_b^a f(t)\ dt = 0.

In complex analysis, the analogous fact is significantly more powerful, and is known as Cauchy’s theorem:

Theorem 4 (Cauchy’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour in {\Omega} (thus {\gamma(1)=\gamma(0)}). Then {\int_\gamma f(z)\ dz = 0}.

Exercise 5 Use Stokes’ theorem to give a proof of Cauchy’s theorem.

A useful reformulation of Cauchy’s theorem is that of contour shifting: if {f: \Omega \rightarrow {\bf C}} is a holomorphic function on a open set {\Omega}, and {\gamma, \tilde \gamma} are two contours in an open set {\Omega} with {\gamma(0)=\tilde \gamma(0)} and {\gamma(1) = \tilde \gamma(1)}, such that {\gamma} can be continuously deformed into {\tilde \gamma}, then {\int_\gamma f(z)\ dz = \int_{\tilde \gamma} f(z)\ dz}. A basic application of contour shifting is the Cauchy integral formula:

Theorem 6 (Cauchy integral formula) Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function in a simply connected open set {\Omega}, and let {\gamma: [0,1] \rightarrow \Omega} be a closed contour which is simple (thus {\gamma} does not traverse any point more than once, with the exception of the endpoint {\gamma(0)=\gamma(1)} that is traversed twice), and which encloses a bounded region {U} in the anticlockwise direction. Then for any {z_0 \in U}, one has

\displaystyle  \int_\gamma \frac{f(z)}{z-z_0}\ dz= 2\pi i f(z_0).

Proof: Let {\varepsilon > 0} be a sufficiently small quantity. By contour shifting, one can replace the contour {\gamma} by the sum (concatenation) of three contours: a contour {\rho} from {\gamma(0)} to {z_0+\varepsilon}, a contour {C_\varepsilon} traversing the circle {\{z: |z-z_0|=\varepsilon\}} once anticlockwise, and the reversal {-\rho} of the contour {\rho} that goes from {z_0+\varepsilon} to {\gamma_0}. The contributions of the contours {\rho, -\rho} cancel each other, thus

\displaystyle \int_\gamma \frac{f(z)}{z-z_0}\ dz = \int_{C_\varepsilon} \frac{f(z)}{z-z_0}\ dz.

By a change of variables, the right-hand side can be expanded as

\displaystyle  2\pi i \int_0^1 f(z_0 + \varepsilon e^{2\pi i t})\ dt.

Sending {\varepsilon \rightarrow 0}, we obtain the claim. \Box

The Cauchy integral formula has many consequences. Specialising to the case when {\gamma} traverses a circle {\{ z: |z-z_0|=r\}} around {z_0}, we conclude the mean value property

\displaystyle  f(z_0) = \int_0^1 f(z_0 + re^{2\pi i t})\ dt \ \ \ \ \ (3)

whenever {f} is holomorphic in a neighbourhood of the disk {\{ z: |z-z_0| \leq r \}}. In a similar spirit, we have the maximum principle for holomorphic functions:

Lemma 7 (Maximum principle) Let {\Omega} be a simply connected open set, and let {\gamma} be a simple closed contour in {\Omega} enclosing a bounded region {U} anti-clockwise. Let {f: \Omega \rightarrow {\bf C}} be a holomorphic function. If we have the bound {|f(z)| \leq M} for all {z} on the contour {\gamma}, then we also have the bound {|f(z_0)| \leq M} for all {z_0 \in U}.

Proof: We use an argument of Landau. Fix {z_0 \in U}. From the Cauchy integral formula and the triangle inequality we have the bound

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma} M

for some constant {C_{z_0,\gamma} > 0} depending on {z_0} and {\gamma}. This ostensibly looks like a weaker bound than what we want, but we can miraculously make the constant {C_{z_0,\gamma}} disappear by the “tensor power trick“. Namely, observe that if {f} is a holomorphic function bounded in magnitude by {M} on {\gamma}, and {n} is a natural number, then {f^n} is a holomorphic function bounded in magnitude by {M^n} on {\gamma}. Applying the preceding argument with {f, M} replaced by {f^n, M^n} we conclude that

\displaystyle  |f(z_0)|^n \leq C_{z_0,\gamma} M^n

and hence

\displaystyle  |f(z_0)| \leq C_{z_0,\gamma}^{1/n} M.

Sending {n \rightarrow \infty}, we obtain the claim. \Box

Another basic application of the integral formula is

Corollary 8 Every holomorphic function {f: \Omega \rightarrow {\bf C}} is complex analytic, thus it has a convergent Taylor series around every point {z_0} in the domain. In particular, holomorphic functions are smooth, and the derivative of a holomorphic function is again holomorphic.

Conversely, it is easy to see that complex analytic functions are holomorphic. Thus, the terms “complex analytic” and “holomorphic” are synonymous, at least when working on open domains. (On a non-open set {\Omega}, saying that {f} is analytic on {\Omega} is equivalent to asserting that {f} extends to a holomorphic function of an open neighbourhood of {\Omega}.) This is in marked contrast to real analysis, in which a function can be continuously differentiable, or even smooth, without being real analytic.

Proof: By translation, we may suppose that {z_0=0}. Let {C_r} be a a contour traversing the circle {\{ z: |z|=r\}} that is contained in the domain {\Omega}, then by the Cauchy integral formula one has

\displaystyle  f(z) = \frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w-z}\ dw

for all {z} in the disk {\{ z: |z| < r \}}. As {f} is continuously differentiable (and hence continuous) on {C_r}, it is bounded. From the geometric series formula

\displaystyle  \frac{1}{w-z} = \frac{1}{w} + \frac{1}{w^2} z + \frac{1}{w^3} z^2 + \dots

and dominated convergence, we conclude that

\displaystyle  f(z) = \sum_{n=0}^\infty (\frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{w^{n+1}}\ dw) z^n

with the right-hand side an absolutely convergent series for {|z| < r}, and the claim follows. \Box

Exercise 9 Establish the generalised Cauchy integral formulae

\displaystyle  f^{(k)}(z_0) = \frac{k!}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{k+1}}\ dz

for any non-negative integer {k}, where {f^{(k)}} is the {k}-fold complex derivative of {f}.

This in turn leads to a converse to Cauchy’s theorem, known as Morera’s theorem:

Corollary 10 (Morera’s theorem) Let {f: \Omega \rightarrow {\bf C}} be a continuous function on an open set {\Omega} with the property that {\int_\gamma f(z)\ dz = 0} for all closed contours {\gamma: [0,1] \rightarrow \Omega}. Then {f} is holomorphic.

Proof: We can of course assume {\Omega} to be non-empty and connected (hence path-connected). Fix a point {z_0 \in \Omega}, and define a “primitive” {F: \Omega \rightarrow {\bf C}} of {f} by defining {F(z_1) = \int_\gamma f(z)\ dz}, with {\gamma: [0,1] \rightarrow \Omega} being any contour from {z_0} to {z_1} (this is well defined by hypothesis). By mimicking the proof of the real fundamental theorem of calculus, we see that {F} is holomorphic with {F'=f}, and the claim now follows from Corollary 8. \Box

An important consequence of Morera’s theorem for us is

Corollary 11 (Locally uniform limit of holomorphic functions is holomorphic) Let {f_n: \Omega \rightarrow {\bf C}} be holomorphic functions on an open set {\Omega} which converge locally uniformly to a function {f: \Omega \rightarrow {\bf C}}. Then {f} is also holomorphic on {\Omega}.

Proof: By working locally we may assume that {\Omega} is a ball, and in particular simply connected. By Cauchy’s theorem, {\int_\gamma f_n(z)\ dz = 0} for all closed contours {\gamma} in {\Omega}. By local uniform convergence, this implies that {\int_\gamma f(z)\ dz = 0} for all such contours, and the claim then follows from Morera’s theorem. \Box

Now we study the zeroes of complex analytic functions. If a complex analytic function {f} vanishes at a point {z_0}, but is not identically zero in a neighbourhood of that point, then by Taylor expansion we see that {f} factors in a sufficiently small neighbourhood of {z_0} as

\displaystyle  f(z) = (z-z_0)^n g(z_0) \ \ \ \ \ (4)

for some natural number {n} (which we call the order or multiplicity of the zero at {f}) and some function {g} that is complex analytic and non-zero near {z_0}; this generalises the factor theorem for polynomials. In particular, the zero {z_0} is isolated if {f} does not vanish identically near {z_0}. We conclude that if {\Omega} is connected and {f} vanishes on a neighbourhood of some point {z_0} in {\Omega}, then it must vanish on all of {\Omega} (since the maximal connected neighbourhood of {z_0} in {\Omega} on which {f} vanishes cannot have any boundary point in {\Omega}). This implies unique continuation of analytic functions: if two complex analytic functions on {\Omega} agree on a non-empty open set, then they agree everywhere. In particular, if a complex analytic function does not vanish everywhere, then all of its zeroes are isolated, so in particular it has only finitely many zeroes on any given compact set.

Recall that a rational function is a function {f} which is a quotient {g/h} of two polynomials (at least outside of the set where {h} vanishes). Analogously, let us define a meromorphic function on an open set {\Omega} to be a function {f: \Omega \backslash S \rightarrow {\bf C}} defined outside of a discrete subset {S} of {\Omega} (the singularities of {f}), which is locally the quotient {g/h} of holomorphic functions, in the sense that for every {z_0 \in \Omega}, one has {f=g/h} in a neighbourhood of {z_0} excluding {S}, with {g, h} holomorphic near {z_0} and with {h} non-vanishing outside of {S}. If {z_0 \in S} and {g} has a zero of equal or higher order than {h} at {z_0}, then the singularity is removable and one can extend the meromorphic function holomorphically across {z_0} (by the holomorphic factor theorem (4)); otherwise, the singularity is non-removable and is known as a pole, whose order is equal to the difference between the order of {h} and the order of {g} at {z_0}. (If one wished, one could extend meromorphic functions to the poles by embedding {{\bf C}} in the Riemann sphere {{\bf C} \cup \{\infty\}} and mapping each pole to {\infty}, but we will not do so here. One could also consider non-meromorphic functions with essential singularities at various points, but we will have no need to analyse such singularities in this course.) If the order of a pole or zero is one, we say that it is simple; if it is two, we say it is double; and so forth.

Exercise 12 Show that the space of meromorphic functions on a non-empty open set {\Omega}, quotiented by almost everywhere equivalence, forms a field.

By quotienting two Taylor series, we see that if a meromorphic function {f} has a pole of order {n} at some point {z_0}, then it has a Laurent expansion

\displaystyle  f = \sum_{m=-n}^\infty a_m (z-z_0)^m,

absolutely convergent in a neighbourhood of {z_0} excluding {z_0} itself, and with {a_{-n}} non-zero. The Laurent coefficient {a_{-1}} has a special significance, and is called the residue of the meromorphic function {f} at {z_0}, which we will denote as {\hbox{Res}(f;z_0)}. The importance of this coefficient comes from the following significant generalisation of the Cauchy integral formula, known as the residue theorem:

Exercise 13 (Residue theorem) Let {f} be a meromorphic function on a simply connected domain {\Omega}, and let {\gamma} be a closed contour in {\Omega} enclosing a bounded region {U} anticlockwise, and avoiding all the singularities of {f}. Show that

\displaystyle  \int_\gamma f(z)\ dz = 2\pi i \sum_\rho \hbox{Res}(f;\rho)

where {\rho} is summed over all the poles of {f} that lie in {U}.

The residue theorem is particularly useful when applied to logarithmic derivatives {f'/f} of meromorphic functions {f}, because the residue is of a specific form:

Exercise 14 Let {f} be a meromorphic function on an open set {\Omega} that does not vanish identically. Show that the only poles of {f'/f} are simple poles (poles of order {1}), occurring at the poles and zeroes of {f} (after all removable singularities have been removed). Furthermore, the residue of {f'/f} at a pole {z_0} is an integer, equal to the order of zero of {f} if {f} has a zero at {z_0}, or equal to negative the order of pole at {f} if {f} has a pole at {z_0}.

Remark 15 The fact that residues of logarithmic derivatives of meromorphic functions are automatically integers is a remarkable feature of the complex analytic approach to multiplicative number theory, which is difficult (though not entirely impossible) to duplicate in other approaches to the subject. Here is a sample application of this integrality, which is challenging to reproduce by non-complex-analytic means: if {f} is meromorphic near {z_0}, and one has the bound {|\frac{f'}{f}(z_0+t)| \leq \frac{0.9}{t} + O(1)} as {t \rightarrow 0^+}, then {\frac{f'}{f}} must in fact stay bounded near {z_0}, because the only integer of magnitude less than {0.9} is zero.

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Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals {{\bf Q}}, and the classical ring of integers {{\bf Z}}, are placed inside the much larger field {\overline{{\bf Q}}} of algebraic numbers, and the much larger ring {{\mathcal A}} of algebraic integers, respectively. Recall that an algebraic number is a root of a polynomial with integer coefficients, and an algebraic integer is a root of a monic polynomial with integer coefficients; thus for instance {\sqrt{2}} is an algebraic integer (a root of {x^2-2}), while {\sqrt{2}/2} is merely an algebraic number (a root of {4x^2-2}). For the purposes of this post, we will adopt the concrete (but somewhat artificial) perspective of viewing algebraic numbers and integers as lying inside the complex numbers {{\bf C}}, thus {{\mathcal A} \subset \overline{{\bf Q}} \subset {\bf C}}. (From a modern algebraic perspective, it is better to think of {\overline{{\bf Q}}} as existing as an abstract field separate from {{\bf C}}, but which has a number of embeddings into {{\bf C}} (as well as into other fields, such as the completed p-adics {{\bf C}_p}), no one of which should be considered favoured over any other; cf. this mathOverflow post. But for the rudimentary algebraic number theory in this post, we will not need to work at this level of abstraction.) In particular, we identify the algebraic integer {\sqrt{-d}} with the complex number {\sqrt{d} i} for any natural number {d}.

Exercise 1 Show that the field of algebraic numbers {\overline{{\bf Q}}} is indeed a field, and that the ring of algebraic integers {{\mathcal A}} is indeed a ring, and is in fact an integral domain. Also, show that {{\bf Z} = {\mathcal A} \cap {\bf Q}}, that is to say the ordinary integers are precisely the algebraic integers that are also rational. Because of this, we will sometimes refer to elements of {{\bf Z}} as rational integers.

In practice, the field {\overline{{\bf Q}}} is too big to conveniently work with directly, having infinite dimension (as a vector space) over {{\bf Q}}. Thus, algebraic number theory generally restricts attention to intermediate fields {{\bf Q} \subset F \subset \overline{{\bf Q}}} between {{\bf Q}} and {\overline{{\bf Q}}}, which are of finite dimension over {{\bf Q}}; that is to say, finite degree extensions of {{\bf Q}}. Such fields are known as algebraic number fields, or number fields for short. Apart from {{\bf Q}} itself, the simplest examples of such number fields are the quadratic fields, which have dimension exactly two over {{\bf Q}}.

Exercise 2 Show that if {\alpha} is a rational number that is not a perfect square, then the field {{\bf Q}(\sqrt{\alpha})} generated by {{\bf Q}} and either of the square roots of {\alpha} is a quadratic field. Conversely, show that all quadratic fields arise in this fashion. (Hint: show that every element of a quadratic field is a root of a quadratic polynomial over the rationals.)

The ring of algebraic integers {{\mathcal A}} is similarly too large to conveniently work with directly, so in algebraic number theory one usually works with the rings {{\mathcal O}_F := {\mathcal A} \cap F} of algebraic integers inside a given number field {F}. One can (and does) study this situation in great generality, but for the purposes of this post we shall restrict attention to a simple but illustrative special case, namely the quadratic fields with a certain type of negative discriminant. (The positive discriminant case will be briefly discussed in Remark 42 below.)

Exercise 3 Let {d} be a square-free natural number with {d=1\ (4)} or {d=2\ (4)}. Show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of algebraic integers in {{\bf Q}(\sqrt{-d})} is given by

\displaystyle  {\mathcal O} = {\bf Z}[\sqrt{-d}] = \{ a + b \sqrt{-d}: a,b \in {\bf Z} \}.

If instead {d} is square-free with {d=3\ (4)}, show that the ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} is instead given by

\displaystyle  {\mathcal O} = {\bf Z}[\frac{1+\sqrt{-d}}{2}] = \{ a + b \frac{1+\sqrt{-d}}{2}: a,b \in {\bf Z} \}.

What happens if {d} is not square-free, or negative?

Remark 4 In the case {d=3\ (4)}, it may naively appear more natural to work with the ring {{\bf Z}[\sqrt{-d}]}, which is an index two subring of {{\mathcal O}}. However, because this ring only captures some of the algebraic integers in {{\bf Q}(\sqrt{-d})} rather than all of them, the algebraic properties of these rings are somewhat worse than those of {{\mathcal O}} (in particular, they generally fail to be Dedekind domains) and so are not convenient to work with in algebraic number theory.

We refer to fields of the form {{\bf Q}(\sqrt{-d})} for natural square-free numbers {d} as quadratic fields of negative discriminant, and similarly refer to {{\mathcal O}_{{\bf Q}(\sqrt{-d})}} as a ring of quadratic integers of negative discriminant. Quadratic fields and quadratic integers of positive discriminant are just as important to analytic number theory as their negative discriminant counterparts, but we will restrict attention to the latter here for simplicity of discussion.

Thus, for instance, when {d=1}, the ring of integers in {{\bf Q}(\sqrt{-1})} is the ring of Gaussian integers

\displaystyle  {\bf Z}[\sqrt{-1}] = \{ x + y \sqrt{-1}: x,y \in {\bf Z} \}

and when {d=3}, the ring of integers in {{\bf Q}(\sqrt{-3})} is the ring of Eisenstein integers

\displaystyle  {\bf Z}[\omega] := \{ x + y \omega: x,y \in {\bf Z} \}

where {\omega := e^{2\pi i /3}} is a cube root of unity.

As these examples illustrate, the additive structure of a ring {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} of quadratic integers is that of a two-dimensional lattice in {{\bf C}}, which is isomorphic as an additive group to {{\bf Z}^2}. Thus, from an additive viewpoint, one can view quadratic integers as “two-dimensional” analogues of rational integers. From a multiplicative viewpoint, however, the quadratic integers (and more generally, integers in a number field) behave very similarly to the rational integers (as opposed to being some sort of “higher-dimensional” version of such integers). Indeed, a large part of basic algebraic number theory is devoted to treating the multiplicative theory of integers in number fields in a unified fashion, that naturally generalises the classical multiplicative theory of the rational integers.

For instance, every rational integer {n \in {\bf Z}} has an absolute value {|n| \in {\bf N} \cup \{0\}}, with the multiplicativity property {|nm| = |n| |m|} for {n,m \in {\bf Z}}, and the positivity property {|n| > 0} for all {n \neq 0}. Among other things, the absolute value detects units: {|n| = 1} if and only if {n} is a unit in {{\bf Z}} (that is to say, it is multiplicatively invertible in {{\bf Z}}). Similarly, in any ring of quadratic integers {{\mathcal O} = {\mathcal O}_{{\bf Q}(\sqrt{-d})}} with negative discriminant, we can assign a norm {N(n) \in {\bf N} \cup \{0\}} to any quadratic integer {n \in {\mathcal O}_{{\bf Q}(\sqrt{-d})}} by the formula

\displaystyle  N(n) = n \overline{n}

where {\overline{n}} is the complex conjugate of {n}. (When working with other number fields than quadratic fields of negative discriminant, one instead defines {N(n)} to be the product of all the Galois conjugates of {n}.) Thus for instance, when {d=1,2\ (4)} one has

\displaystyle  N(x + y \sqrt{-d}) = x^2 + dy^2 \ \ \ \ \ (1)

and when {d=3\ (4)} one has

\displaystyle  N(x + y \frac{1+\sqrt{-d}}{2}) = x^2 + xy + \frac{d+1}{4} y^2. \ \ \ \ \ (2)

Analogously to the rational integers, we have the multiplicativity property {N(nm) = N(n) N(m)} for {n,m \in {\mathcal O}} and the positivity property {N(n) > 0} for {n \neq 0}, and the units in {{\mathcal O}} are precisely the elements of norm one.

Exercise 5 Establish the three claims of the previous paragraph. Conclude that the units (invertible elements) of {{\mathcal O}} consist of the four elements {\pm 1, \pm i} if {d=1}, the six elements {\pm 1, \pm \omega, \pm \omega^2} if {d=3}, and the two elements {\pm 1} if {d \neq 1,3}.

For the rational integers, we of course have the fundamental theorem of arithmetic, which asserts that every non-zero rational integer can be uniquely factored (up to permutation and units) as the product of irreducible integers, that is to say non-zero, non-unit integers that cannot be factored into the product of integers of strictly smaller norm. As it turns out, the same claim is true for a few additional rings of quadratic integers, such as the Gaussian integers and Eisenstein integers, but fails in general; for instance, in the ring {{\bf Z}[\sqrt{-5}]}, we have the famous counterexample

\displaystyle  6 = 2 \times 3 = (1+\sqrt{-5}) (1-\sqrt{-5})

that decomposes {6} non-uniquely into the product of irreducibles in {{\bf Z}[\sqrt{-5}]}. Nevertheless, it is an important fact that the fundamental theorem of arithmetic can be salvaged if one uses an “idealised” notion of a number in a ring of integers {{\mathcal O}}, now known in modern language as an ideal of that ring. For instance, in {{\bf Z}[\sqrt{-5}]}, the principal ideal {(6)} turns out to uniquely factor into the product of (non-principal) ideals {(2) + (1+\sqrt{-5}), (2) + (1-\sqrt{-5}), (3) + (1+\sqrt{-5}), (3) + (1-\sqrt{-5})}; see Exercise 27. We will review the basic theory of ideals in number fields (focusing primarily on quadratic fields of negative discriminant) below the fold.

The norm forms (1), (2) can be viewed as examples of positive definite quadratic forms {Q: {\bf Z}^2 \rightarrow {\bf Z}} over the integers, by which we mean a polynomial of the form

\displaystyle  Q(x,y) = ax^2 + bxy + cy^2

for some integer coefficients {a,b,c}. One can declare two quadratic forms {Q, Q': {\bf Z}^2 \rightarrow {\bf Z}} to be equivalent if one can transform one to the other by an invertible linear transformation {T: {\bf Z}^2 \rightarrow {\bf Z}^2}, so that {Q' = Q \circ T}. For example, the quadratic forms {(x,y) \mapsto x^2 + y^2} and {(x',y') \mapsto 2 (x')^2 + 2 x' y' + (y')^2} are equivalent, as can be seen by using the invertible linear transformation {(x,y) = (x',x'+y')}. Such equivalences correspond to the different choices of basis available when expressing a ring such as {{\mathcal O}} (or an ideal thereof) additively as a copy of {{\bf Z}^2}.

There is an important and classical invariant of a quadratic form {(x,y) \mapsto ax^2 + bxy + c y^2}, namely the discriminant {\Delta := b^2 - 4ac}, which will of course be familiar to most readers via the quadratic formula, which among other things tells us that a quadratic form will be positive definite precisely when its discriminant is negative. It is not difficult (particularly if one exploits the multiplicativity of the determinant of {2 \times 2} matrices) to show that two equivalent quadratic forms have the same discriminant. Thus for instance any quadratic form equivalent to (1) has discriminant {-4d}, while any quadratic form equivalent to (2) has discriminant {-d}. Thus we see that each ring {{\mathcal O}[\sqrt{-d}]} of quadratic integers is associated with a certain negative discriminant {D}, defined to equal {-4d} when {d=1,2\ (4)} and {-d} when {d=3\ (4)}.

Exercise 6 (Geometric interpretation of discriminant) Let {Q: {\bf Z}^2 \rightarrow {\bf Z}} be a quadratic form of negative discriminant {D}, and extend it to a real form {Q: {\bf R}^2 \rightarrow {\bf R}} in the obvious fashion. Show that for any {X>0}, the set {\{ (x,y) \in {\bf R}^2: Q(x,y) \leq X \}} is an ellipse of area {2\pi X / \sqrt{|D|}}.

It is natural to ask the converse question: if two quadratic forms have the same discriminant, are they necessarily equivalent? For certain choices of discriminant, this is the case:

Exercise 7 Show that any quadratic form {ax^2+bxy+cy^2} of discriminant {-4} is equivalent to the form {x^2+y^2}, and any quadratic form of discriminant {-3} is equivalent to {x^2+xy+y^2}. (Hint: use elementary transformations to try to make {|b|} as small as possible, to the point where one only has to check a finite number of cases; this argument is due to Legendre.) More generally, show that for any negative discriminant {D}, there are only finitely many quadratic forms of that discriminant up to equivalence (a result first established by Gauss).

Unfortunately, for most choices of discriminant, the converse question fails; for instance, the quadratic forms {x^2+5y^2} and {2x^2+2xy+3y^2} both have discriminant {-20}, but are not equivalent (Exercise 38). This particular failure of equivalence turns out to be intimately related to the failure of unique factorisation in the ring {{\bf Z}[\sqrt{-5}]}.

It turns out that there is a fundamental connection between quadratic fields, equivalence classes of quadratic forms of a given discriminant, and real Dirichlet characters, thus connecting the material discussed above with the last section of the previous set of notes. Here is a typical instance of this connection:

Proposition 8 Let {\chi_4: {\bf N} \rightarrow {\bf R}} be the real non-principal Dirichlet character of modulus {4}, or more explicitly {\chi_4(n)} is equal to {+1} when {n = 1\ (4)}, {-1} when {n = 3\ (4)}, and {0} when {n = 0,2\ (4)}.

  • (i) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} with norm {N(m)=n} is equal to {4(1 * \chi_4)(n)}. Equivalently, the number of solutions to the equation {n = x^2+y^2} with {x,y \in{\bf Z}} is {4(1*\chi_4)(n)}. (Here, as in the previous post, the symbol {*} denotes Dirichlet convolution.)
  • (ii) For any natural number {n}, the number of Gaussian integers {m \in {\bf Z}[\sqrt{-1}]} that divide {n} (thus {n = dm} for some {d \in {\bf Z}[\sqrt{-1}]}) is {4(1*1*1*\mu\chi_4)(n)}.

We will prove this proposition later in these notes. We observe that as a special case of part (i) of this proposition, we recover the Fermat two-square theorem: an odd prime {p} is expressible as the sum of two squares if and only if {p = 1\ (4)}. This proposition should also be compared with the fact, used crucially in the previous post to prove Dirichlet’s theorem, that {1*\chi(n)} is non-negative for any {n}, and at least one when {n} is a square, for any quadratic character {\chi}.

As an illustration of the relevance of such connections to analytic number theory, let us now explicitly compute {L(1,\chi_4)}.

Corollary 9 {L(1,\chi_4) = \frac{\pi}{4}}.

This particular identity is also known as the Leibniz formula.

Proof: For a large number {x}, consider the quantity

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1

of all the Gaussian integers of norm less than {x}. On the one hand, this is the same as the number of lattice points of {{\bf Z}^2} in the disk {\{ (a,b) \in {\bf R}^2: a^2+b^2 \leq x \}} of radius {\sqrt{x}}. Placing a unit square centred at each such lattice point, we obtain a region which differs from the disk by a region contained in an annulus of area {O(\sqrt{x})}. As the area of the disk is {\pi x}, we conclude the Gauss bound

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = \pi x + O(\sqrt{x}).

On the other hand, by Proposition 8(i) (and removing the {n=0} contribution), we see that

\displaystyle  \sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1 = 1 + 4 \sum_{n \leq x} 1 * \chi_4(n).

Now we use the Dirichlet hyperbola method to expand the right-hand side sum, first expressing

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = \sum_{d \leq \sqrt{x}} \chi_4(d) \sum_{m \leq x/d} 1 + \sum_{m \leq \sqrt{x}} \sum_{d \leq x/m} \chi_4(d)

\displaystyle  - (\sum_{d \leq \sqrt{x}} \chi_4(d)) (\sum_{m \leq \sqrt{x}} 1)

and then using the bounds {\sum_{d \leq y} \chi_4(d) = O(1)}, {\sum_{m \leq y} 1 = y + O(1)}, {\sum_{d \leq \sqrt{x}} \frac{\chi_4(d)}{d} = L(1,\chi_4) + O(\frac{1}{\sqrt{x}})} from the previous set of notes to conclude that

\displaystyle  \sum_{n \leq x} 1 * \chi_4(n) = x L(1,\chi_4) + O(\sqrt{x}).

Comparing the two formulae for {\sum_{n \in {\bf Z}[\sqrt{-1}]: N(n) \leq x} 1} and sending {x \rightarrow \infty}, we obtain the claim. \Box

Exercise 10 Give an alternate proof of Corollary 9 that relies on obtaining asymptotics for the Dirichlet series {\sum_{n \in {\bf Z}} \frac{1 * \chi_4(n)}{n^s}} as {s \rightarrow 1^+}, rather than using the Dirichlet hyperbola method.

Exercise 11 Give a direct proof of Corollary 9 that does not use Proposition 8, instead using Taylor expansion of the complex logarithm {\log(1+z)}. (One can also use Taylor expansions of some other functions related to the complex logarithm here, such as the arctangent function.)

More generally, one can relate {L(1,\chi)} for a real Dirichlet character {\chi} with the number of inequivalent quadratic forms of a certain discriminant, via the famous class number formula; we will give a special case of this formula below the fold.

The material here is only a very rudimentary introduction to algebraic number theory, and is not essential to the rest of the course. A slightly expanded version of the material here, from the perspective of analytic number theory, may be found in Sections 5 and 6 of Davenport’s book. A more in-depth treatment of algebraic number theory may be found in a number of texts, e.g. Fröhlich and Taylor.

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